PHYSICAL  CHEMISTRY 


EWELL 


A  TEXT-BOOK  OF 

PHYSICAL  CHEMISTRY 
THEORY   AND    PRACTICE 


BY 

ARTHUR  W.  EWELL,  Ph.  D. 

i  \ 

ASSISTANT   PROFESSOR    OF    PHYSICS,    WORCESTER    POLYTECHNIC    INSTITUTE 


WITH  ONE  HUNDRED  AND  TWO  ILLUSTRATIONS 
AND   SIXTY-THREE   TABLES 


k-***- 

•  F  THE 

UNIVERSITY 

OF 


PHILADELPHIA 
P.  BLAKISTON'S  SON  &  CO. 

1012  WALNUT   STREET 
1909 


GENERAL 

c 


COPYRIGHT,  1909,  BY  P.  BLAKISTON'S  SON  &  Co. 


Printed  by 

The  Maple  Press 

York,  Pa. 


PREFACE. 


This  book  is  intended  to  serve  as  a  laboratory  manual,  as  a 
text-book  to  accompany  recitations  or  lectures  and  as  a  conven- 
ient book  of  reference.  The  author  has  felt  the  need  of  such 
a  book  in  a  general  course  in  Physical  Chemistry  which  he  has 
conducted  for  several  years,  but  has  failed  to  find  any  single 
book  of  this  scope.  This  book  has  been  prepared  in  the  belief 
that  others  have  felt  a  similar  want. 

It  is  designed  for  students  in  American  colleges  and  tech- 
nical schools  who  have  completed  the  equivalent  of  the 
prescribed  Freshman  and  Sophomore  class-room  and  laboratory 
courses,  in  mathematics,  physics,  and  chemistry.  A  knowledge 
of  the  calculus  is  assumed  in  many  of  the  theoretical  discus- 
sions. The  value  and  necessity  of  the  calculus  have  been  so 
emphasized  during  recent  years  that  most  students  of  this 
grade  have  studied  it.  The  paragraphing  has,  however,  been 
arranged  in  such  a  manner  that  students  unfamiliar  with 
the  calculus  can  omit  the  portions  in  which  it  is  employed 
assume  the  results  of  the  omitted  discussions  or  derivations. 

The  laboratory  exercises  are  chosen  with  the  aim  of  giving 
the  student  a 'clear  understanding  of  the  principles  involved  in 
the  subjects  usually  included  under  the  title  of  Physical  Chem- 
istry, and  also  in  certain  other  subjects  in  advanced  physics, 
which,  though  not  usually  included  under  this  title,  are  of  par- 
ticular importance  to  chemists. 

Effort  has  been  made  to  have  the  experiments  more  than 
mere  manipulation.  Close  arid  careful  thinking  is  required 
in  working  up  many  of  the  experiments,  the  results  of  practi- 
cally all  of  which  are  compared  with  the  results  required  by  the 
theory  immediately  preceding.  Each  experiment  is  followed 
by^questions  designed  to  stimulate  thought  upon  the  principles 

id  applications  of  the  experiment.     All  but  a  very  few  of  the 


187211 


VI  PREFACE. 

exercises  described  have  been  performed  by  the  author's 
laboratory  class  for  several  years.  The  few  exceptions  have 
been  tested  by  the  author  under  laboratory  conditions. 

It  is  probable  that  the  majority  of  students  will  have  had 
already  a  few  of  the  more  distinctly  physical  experiments. 
Much  of  the  general  information  regarding  units,  etc.,  in  the 
Introduction  and  in  the  first  part  of  several  chapters  is  also 
probably  familiar  to  some,  but  is  included  for  completeness 
and  for  reference. 

Physical  Chemistry  apparatus  which  is  understood  with 
difficulty  or  the  description  of  which  is  instructive,  is  described 
under  the  appropriate  topic  in  the  body  of  the  book.  All 
other  apparatus,  methods,  etc.,  are  described  in  the  Intro- 
duction. The  Introduction  also  contains  references  to  all  the 
apparatus  considered  elsewhere,  together  with  mathematical 
data,  discussion  of  errors,  etc.,  so  that  the  book  constitutes  a 
complete  manual  for  all  ordinary  work  in  Physical  Chemistry. 

Experiments  requiring  expensive  chemicals  are  avoided  and 
apparatus  which  is  not  readily  procurable  is  not  employed. 
Any  apparatus  which  cannot  readily  be  purchased  can  be 
constructed  easily  from  the  accompanying  directions.  Sug- 
gestions are  given  with  each  experiment  for  suitable  materials, 
etc.,  and  in  almost  every  one  there  is  a  choice  of  several  sub- 
stances, so  that  there  may  be  some  variation  in  each  experi- 
ment as  performed  by  different  students. 

Theory  and  experiment  are  logically  arranged  together.  It 
is  usually  impossible  or  undesirable  to  have  all  the  students  in 
such  a  course  pursuing  the  same  exercise,  and  therefore  many 
of  the  students  will  necessarily  be  working  in  advance  of  the 
text  if  the  laboratory  work  and  the  recitations  are  parallel. 
Each  experiment,  however,  is  immediately  preceded  by  its 
theory.  The  student  may  have  to  defer  a  complete  grasp  of 
the  experiment  until  he  has  reached  the  corresponding  point 
in  the  text. 

A  large  number  of  problems  covering  every  topic  are  distrib- 
uted throughout  the  book.  If  these  are  carefully  considered 
and  the  attempts  at  solution  are  written  out  and  handed  in  to 


PREFACE.  Vll 

the  instructor,  it  will  contribute  an  important  factor  toward 
the  attainment  of  the  goal  of  such  a  course — the  ability  to 
discover  problems  in  Physical  Chemistry,  to  meet  such  problems 
with  the  best  mental  and  mechanical  methods,  and  to  find 
pleasure  in  the  struggle. 

Mathematical  tables  and  tables  constantly  employed 
throughout  the  book  are  placed  at  the  end.  With  these 
tables  are  references,  arranged  by  topics,  to  all  tables  in  the 
Introduction  and  main  portion  of  the  book,  so  that  the  most 
recent  determinations  of  any  quantity  employed  in  Physical 
Chemistry  may  be  found  by  reference  to  the  final  section  of 
the  book. 

The  figures  are  schematic  rather  than  detailed,  in  order  to 
bring  out  more  clearly  what  is  essential.  Therefore  different 
scales  are  often  used  for  different  portions  (e.g.,  subsidiary 
electrical  connections  are  often  on  a  much  smaller  scale  than 
the  main  apparatus) . 

The  author  has  attempted  to  describe  the  most  important 
facts  of  Physical  Chemistry  and  to  derive  and  develop  as 
fully  as  possible  the  fundamental  principles.  Details  regard- 
ing minor  extensions  or  exceptions  will  be  found  in  the  numer- 
ous references  in  the  foot-notes. 

The  writer  wishes  to  acknowledge  his  indebtedness  to  the 
following  standard  works:  Nernst,  Theoretical  Chemistry; 
Van't  Hoff's  Lectures;  Jiiptner's  Physikalische  Chemie; 
Reychler-Kuhn,  Physikalisch-Chemische  Theorieen;  Walker's 
Introduction  to  Physical  Chemistry,  and  Findlay's  Practical 
Physical  Chemistry. 

Dr.  A.  Wilmer  Duff  kindly  examined  certain  portions  of  the 
manuscript  and  the  undersigned  is  indebted  to  him  for  many 
valuable  suggestions  and  criticisms. 

ARTHUR  W.  EWELL. 

WORCESTER,  September,   1909. 


CONTENTS. 


PAGE 

INTRODUCTION i 

CHAPTER  I. 
GASES,  VAPORS  AND  LIQUIDS 79 

CHAPTER  II. 
THERMODYNAMICS ...    .    .    .    .    .*  .    .    123 

CHAPTER  III. 
SOLUTIONS T> .    .    .-*•.'...    150 

CHAPTER  IV. 
THERMOCHEMISTRY 181 

CHAPTER  V. 
LIGHT '.    .    .    ...    .    ....    .    .    198 

CHAPTER  VI. 
CHEMICAL  KINETICS .    .    .'  .    .    225 

CHAPTER  VII. 
CHEMICAL  STATICS 250 

CHAPTER  VIII. 
ELECTROLYTIC  CONDUCTION 279 

CHAPTER  IX. 
POTENTIAL  DIFFERENCES 313 

CHAPTER  X. 
GASEOUS  IONS,  RADIOACTIVITY 339 

TABLES 347 

ix 


HE     " 

DIVERSITY 

OF 


PHYSICAL  CHEMISTRY. 


INTRODUCTION. 


i.  Symbols. — Table  I  gives  the  symbols  of  all  the  quantities 
which  frequently  appear  in  this  book.  Customary  usage  has 
been  followed  as  far  as  possible  without  confusing  duplication 
in  the  use  of  symbols.  Quantities  which  appear  under  one 
topic  only  are  not  listed  in  this  table  and  are  usually  rep- 
resented by  the  symbols  used  by  the  original  investigators. 


TABLE  I. 


Symbols. 


temp.  coef. 
ist  constant  of 
,  van  der  Waal's  Eq. 
B  =  radiant  energy 
b  =2d  const,  of 

van.  d.  Waal's  Eq. 
C  \  =  f  constant 
c  j        \  concentration 
D  =Coef.  of  diffusion 
,  =   /  distance 

I  sign  of  differential 
f  difference  of  potential 
E  =  <  electromotive  force 

[  coefficient  of  elasticity 
'  Naperian  base 
small  quantity 
electric  charge 
water  equivalent 
F  =  fluidity 
G  =  additive  property 
g  =  acceleration  of  gravity 
H  =  kinetic    energy    of    mole- 
cules 

h  =  height 
/  =  internal  energy 
i  =  electric  current 
/  =  mechanical  equivalent 
*/  =  ionic  conductivity 
•  iissociation  constant 


k  = 


I 

M 
m 

N 

n 
P 

a 

q 


f  molecular  elevation 

j  (molecular  lowering  =&') 

I  velocity  constant 

[  specific  conductivity 
=  latent  heat  per  gram  mo  le- 

cule 

=  latent  heat  per  gram 
=  molecular  weight 
=  mass  (grams) 
=  no.  atoms  in  molecule 
=  /number 

\  transport  number 

f  number 

\  normal 

>  =  pressure 

=  quantity,  heat    per   gram 

molecule 

=  quantity  of  heat  per  gram 
f  gas  constant  for  gram 

molecule 

=  <  (gas    constant    for    one 
gram  =  R') 
resistance 
=  specific    heat    per    gram 

molecule 
=  specific  heat  per  gram 

f  titration 
=  |  time 

I  surface  tension 


INTRODUCTION. 


,  =   /  temperature 

\  in  Chap.  VI,  =time 

u  \  =  velocity 

v\  =  volume 

W  =work,  energy 
w  =  number    of    gram    mole- 
cules 

X  =  /  f  orce 

'  1  unknown  quantity 
f  unknown  quantity 
\  change  in  concentration 
y  =  fraction 
Z  =  coefficient  of  solubility 

f  temp,     coef .    de- 
ft (alpha)  =  (  gree   of   dissocia- 
[      tion 

2d      power      con- 
stant in  thermal 


(beta)  = 

equations 
(gamma)  =  ratio    of    specific 
heats 


J  (delta)  =  prefix,  signifying 
that  the  quan- 
tity is  small 

e  (epsilon)  =  dielectric  constant 
r?  (eta)  =coef.  of  viscosity 
6  (theta)  =  absolute     temper- 
ature 

f  wave  length 
"j  equivalent  con- 
A  (lambda)  =  \       ductivity 

reduced      pres- 

sure 

,       N         f  refractive    index 
''  (mu)=i  reduced  volume 

f      x       f  reduced  temperature 
v  <nu)  *  I  valency 
p  (rho)  =  density 

fspecific    resist- 

a  (siSma>  =  \  radfatlon     con- 

[     stant 
0  (phi)  =  angle 


2.  NOTATION  OF  VERY  LARGE  AND  VERY  SMALL 
NUMBERS. 

Partly  to  save  space  and  partly  to  indicate  at  once  the 
magnitude  of  very  large  or  very  small  numbers,  the  following 
notation  is  often  used.  The  digits  are  written  down  and  a 
decimal  point  placed  after  the  first,  and  its  position  in  the 
scale  indicated  by  multiplying  by  some  power  of  10.  Thus 
42140000  is  written  4.214X107  and  .00000588  is  written 
5-88Xio~6.  This  also  enables  us  to  abbreviate  the  multi- 
plication and  division  of  such  numbers.  Thus,  421 40000  X 
.00000588  is  the  same  as  4.214X5.88X10  and  42140000-=- 
.00000588  is  the  same  as  (4.124-1-5.88)  Xio13. 


UNITS. 

3.  Fundamental  Units. — Unless  otherwise  specified,  the 
unit  of  length  throughout  this  book  is  the  centimeter,  the 
unit  of  mass  is  the  gram,  and  the  unit  of  time  is  the  second. 


UNITS.  3 

In  measuring  the  wave  length  of  ether  waves  the  following 
units  are  often  employed : 

^(=0.00 1  mm. 

micron  =/*//  =  .000001  mm. 

Angstrom  unit  =  .0000001  mm. 

In  specifying  the  concentration  of  solutions  and  also  in 
problems  concerning  gases,  the  gram-molecule  is  the  unit  of 
mass;  that  is,  in  place  of  one  gram,  the  unit  is  a  number  of 
grams  numerically  equal  to  the  molecular  weight.  For 
example,  the  unit  for  sodium  chloride  is  58.5  grams,  for 
oxygen  the  unit  is  32  grams.  Sometimes  the  gram-atom  is  a 
convenient  unit  of  mass.  A  gram  atom  of  oxygen  would  be 
1 6  grams,  of  chlorine  35  grams,  etc. 

If  the  valency  of  the  dissolved  substance  is  greater  than  one, 
the  gram  equivalent  is  usually  chosen  as  the  unit;  that  is,  a 
number  of  grams  equal  to  the  molecular  or  atomic  weight 
divided  by  the  valency.  A  normal  solution  contains  one  gram 
equivalent  of  the  dissolved  substance  in  one  litre  of  the  solu- 
tion. A  litre  of  such  a  solution  contains,  therefore,,  58.5  grams 
of  sodium  chloride,  or  63.4  grams  of  ferrous  chloride,  or  54.18 
grams  of  ferric  chloride  or  31.8  grams  of  copper,  etc.  The 
concentration  of  a  solution  in  terms  of  that  of  a  normal 
solution  is  designated  by  a  fraction  preceding  the  letter  n. 
Thus  .02n  NaCl  signifies  a  one-fiftieth  normal  solution  of 
salt,  etc. 

The  density  of  a  body  is  the  mass  divided  by  the  volume, 
or,  if  the  density  is  constant,  the  mass  per  unit  volume. 

4.  A  body  of  gas  or  vapor  is  said  to  be  under  standard 
conditions  if  its  temperature  is  o°  and  if  its  pressure  is  equal 
to  that  of  a  column  of  o°  mercury,  76  cm.  high.  The  stand- 
ard force  of  gravity  for  estimating  this  pressure  is  that  at 
sea  level,  and  45°  latitude  (980.6,  §41).  Equations  for  calcu- 
lating the  volume  or  density  under  standard  conditions  are 
given  in  §103. 


4  INTRODUCTION. 

MECHANICAL  UNITS. 

5.  (Velocity)  =  i    cm.   per   sec.     (Acceleration)  =i    cm.   per 
sec.  per  sec.     (Force)  =(dyne)  =  force  required  to  give  a  mass 
of  one  gram  an  acceleration  of  i  cm.  per  sec.  per  sec. 

(Pressure)  =  force  -=-  area.  The  absolute  unit  of  pressure  is 
one  dyne  per  square  centimeter.  A  more  customary  unit  is  the 
weight  of  one  cubic  centimeter  of  mercury  per  square  centimeter 
(reduced  to  o°,  sea  level,  and  45°  latitude)  =13.59X980.6  or 
13326  dynes  per  sq.  cm. 

Energy  and  Work. — The  absolute  unit  is  the  erg=the  work 
done  by  one  dyne  acting  through  a  distance  of  i  cm.  This 
unit  is  very  small  and  a  more  practical  unit  is  the  joule  =  io7 
ergs. 

Activity,  Power,  or  Rate  of  Doing  Work. — The  watt  is  one 
joule  per  second.  1000  joules  per  second  is  called  a  kilowatt. 
One  horse  power  is  746  watts. 

HEAT  UNITS. 

6.  Temperature. — The  centigrade  unit  of  temperature  is  one- 
hundredth  of  the  increase  of  pressure  of  a  perfect  gas,  between 
ice-water  and   steam  at  76  cm.  pressure,  the  volume  being 
constant  (§§101,  149). 

The  Fahrenheit  unit  of  temperature  is  5  /  9  of  the  centigrade 
unit  and  the  temperature  of  ice  and  water  is  called  3  2  instead 
of  zero  as  in  the  centigrade  scale. 

7.  The  unit  of  heat  energy  is  the  calorie  and  is  the  amount 
of  heat  absorbed  by  one  gram  of  water  in  warming  one  degree 
centigrade.     The  most  common  range  is  15°  to  16°  and  this 
calorie  is  equal  to  4.i87Xio7  ergs  =4. 187  joules.     The  large 
calorie  is  the  amount  of  heat  required  to  raise  the  temperature 
of  one  kilogram  of  water  one  degree.     Table  II  illustrates  how 
the   amount  of  heat  required  to  raise  the  temperature  of  a 
gram  of  water  one  degree  varies  with  the  temperature. 

The  unit  of  heat  energy  commonly  employed  in  engineering 
work  in  England  and  America  is  the  British  Thermal  Unit 
-which  rs  usually  abbreviated  to  B.T.U.  i-B.T.U.  =453.6X5/9, 


ELECTRICAL    UNITS.  5 

or  252  calories.  Heat  values,  calorific  powers,  or  heats  of 
combustion  are  often  expressed  in  calories  per  gram,  or  B.T.U. 
per  pound.  Evidently  the  unit  of  mass  cancels  out,  and, 
therefore,  one  calorie  per  gram  is  9/5  of  one  B.T.U.  per  pound.' 
For  example,  1000  calories  =3. 97  B.T.U.'s.  A  heat  value 
of  1000  calories  per  gram  is  equal  to  1800  B.T.U.'s  per  pound. 

TABLE  II. 

Specific  Heat  of  Water. — 5  is  the  true  specific  heat  at  temperature  t, 
in  terms  of  the  o° — 1°  calorie.  (s)  is  the  similar  mean  specific  heat 
between  o°  and  t°.  To  change  to  the  15° — 16°  calorie,  multiply  by 
1.0075. 


t 

> 

(*) 

o 

i  .0000 

i  .0000 

5 

0.9967 

0.9983 

10 

0.9942 

0.9967 

15 

0.9925 

0.9956 

20 

o  .9916 

0.9947 

25 

0.9914 

0.9941 

3° 

0-99*5 

0.9936 

35 

o  .9922 

0.9934 

40 

o  9933 

0-9933 

45 

o  .9946 

0-9933 

5° 

o  .9962 

0-9935 

55 

0.9929 

0.9938 

60 

°-9995 

0.9942 

65 

.0010 

0.9946 

70 

.0025 

0.9952 

75 

.0039 

0.9958 

80 

.0047 

0.9963 

85 

•0053 

o  .9968 

90 

.0052 

o-9973 

95 

1.0045 

0.9976 

IOO 

1.0033 

0.9979 

ELECTRICAL  UNITS. 

8.  Absolute  Electromagnetic  Units. — A  Unit  Magnetic 
Pole  is  a  pole  which,  placed  one  centimeter  from  a  similar 
pole,  is  repelled  with  a  force  of  one  dyne.  The  magnetic 
force  Sit  a  point  is  measured  by  the  force  (in  dynes)  upon  a 
unit  positive  pole. 


6  INTRODUCTION. 

The  absolute  unit  of  current  is  the  current  which,  flowing 
through  a  wire  one  centimeter  long,  bent  into  a  circular  arc 
of  one  centimeter  radius,  produces  unit  magnetic  force  at 
the  center. 

The  unit  of  charge  or  quantity  of  electricity  is  the  amount 
transported  by  unit  current  in  one  second. 

The  difference  of  potential,  or,  the  electromotive  force  between 
two  bodies  or  two  points,  is  measured,  in  absolute  units, 
by  the  work,  in  ergs,  required  to  carry  a  unit  positive  charge 
of  electricity  from  one  to  the  other. 

A  conductor  has  unit  resistance,  if  unit  difference  of  potential 
at  the  terminals  produces  unit  current. 

9.  Absolute  Electrostatic  Units.  —  We  may  also  define  a 
unit  charge  of  electricity  analogously  to  the  above  definition  of 
unit  pole.  A  body  has  unit  charge  if,  placed  one  centimeter 
from  a  similarly  charged  body,  it  is  repelled  with  a  force  of  one 
dyne.  The  electromagnetic  unit  of  charge  is  3  Xio10  (velocity 
of  light)  greater  than  the  electrostatic  unit.  The  electric  force 
between  two  bodies  having  charges  e^  and  e2  is  given  by  the 
expression 


Where  d  is  the  distance  in  centimeters  between  the  charges,  e 
is  a  constant  called  the  dielectric  constant  (Table  XLIII),  and 
F  is  expressed  in  dynes. 

The  difference  of  potential  or  electromotive  force  between  two 
bodies  or  two  points  is  one  electrostatic  unit,  if  one  erg  of 
work  is  required  to  convey  an  electrostatic  unit  of'  electricity 
from  one  to  the  other. 

The  capacity  of  a  body  is  the  charge  required  to  change 
its  potential  by  unity,  or  the  ratio  of  the  charge  to  the 
potential.  If  both  charge  and  potential  are  expressed  in 
electrostatic  units,  the  capacity  of  a  condenser  consisting 
of  parallel  plates  whose  area  is  A,  separated  by  d  centimeters 
of  a  dielectric  whose  constant  is  e,  is 

~     As 

C  =  —  j  (2) 

47T  d 


ELECTRICAL    UNITS.  7 

The  Electrical  Energy,  in  ergs,  of  a  body  having  a  charge  e 
and  a  potential  E  is 

W  =  \Ee  (3) 

10.  Practical  Units. — The  above  units  give  either  incon- 
veniently large  or  inconveniently  small  numbers  for  the 
electrical  quantities  commonly  used,  and,  therefore,  multiples 
or  submultiples  are  used  as  practical  units. 

The  practical  unit  of  current  is  the  ampere,  which  is  one- 
tenth  of  the  electromagnetic  unit.  The  practical  unit  of 
electric  charge  or  quantity  is  the  charge  carried  by  one  ampere 
in  one  second,  and  is  called  a  coulomb.  The  coulomb  is  evi- 
dently, also,  one-tenth  of  the  corresponding  electromagnetic 
unit. 

The  practical  unit  of  difference  of  potential  or  electromotive 
force  is  the  volt.  The  potential  difference  between  two  bodies 
or  two  points  is  one  volt  if  one  practical  unit  of  work,  the 
joule,  is  required  to  carry  one  coulomb  from  one  point  to  the 
other.  Since  one  joule  is  io7  absolute  units  (ergs)  and  one 
coulomb  is  io~x,  the  volt  is  io7  -j-io-1  =io8  electromagnetic 
units  of  potential. 

The  quantity  of  electricity  carried  by  a  current  of  i  amperes, 
flowing  for  T  seconds,  is  i  T  coulombs,  and  if  the  difference  of 
potential  of  the  two  points  between  which  it  flows  is  E,  the 
work  done  between  these  points  is 

E  i  t 

W  =  E  i  T  joules  =  -calories  (4) 

4- 187 

The  rate  at  which  work  is  done  is 

W      -r  .  joules         _  . 

-^=E  i  —      -  =  E  i  watts  (c) 

T  sec 

or  the  product  of  amperes   by  volts  gives  the  electric  power  in 
watts. 

The  practical  unit  of  resistance  is  the  ohm  and  is  equal 
to  the  resistance  of  a  conductor  which  carries  one  ampere  when 
the  difference  of  potential  of  the  ends  is  one  volt.  Since 


8 


INTRODUCTION. 


the  resistance  is  equal  to  the  difference  of  potential  divided 
by  the  current,  one  ohm  is  equal  to  io8-Mo~I==io9  electro- 
magnetic units.  The  resistance  of  a  column  of  mercury 
106.3  cm-  l°n£  and  one  square  millimeter  cross  section,  at  o°, 
is  one  ohm. 

The  resistance  of  a  conductor  may  be  expressed  in  terms  of 
the  length  /,  the  cross  section,  A,  and  a  constant,  <r,  called  the 
specific  resistance 


K-.i 


(6) 


Evidently  a  may  be  defined  as  the  resistance  of  a  centimeter 
cube  of  the  material  of  this  conductor. 

The  reciprocal  of  the  resistance  of  a  conductor  is  called 
the  conductance.  The  specific  conductivity,  k,  is  the  reciprocal 
of  the  specific  resistance 


'AR 


(7) 


The  equivalent  and  molecular  conductivities  are  defined  in 
§289. 

The  practical  unit  of  capacity  is  the  farad.  A  condenser 
has  a  capacity  of  one  farad,  if  the  potential  difference  of  its 
plates  is  one  volt  when  it  is  charged  with  one  coulomb.  A 
farad  is  equal  to  9X10"  electrostatic  units  of  capacity. 

TABLE  III. 

Conversion  Factors  for  Electrical  Units. 


Unit 

Practical 

Electromagnetic 

Electrostatic 

Coulomb                   •  •  • 

' 

i 

IO-1 

'  3  X  i  o9 

Ampere  
Volt  
Ohm               

i 

! 

io-7 
i  o» 

109 

3  XIOP 
3.33X10-3 

Farad  .  . 

i 

10-9 

0  XlO11 

Multiples  and  Submultiples. — The  prefix  kilo-  before  a  unit 
signifies  a  unit  one  thousand  times  greater;  mega-,  a  unit  one 


LOGARITHMS.  .  9 

million  times  greater.  Milli-,  signifies  a  unit  one  thousandth 
times  as  great,  and  micro-,  a  unit  one  million  times  smaller. 
For  example,  2  kilo-volts  =  2000  volts.  3  milliamperes  =  .003 
amperes.  4  microfarads  =  .000004  farads,  etc. 

12.  Table    III    gives   the    factors   by   which   the   numbers 
representing  different  electric  quantities,  expressed  in  practi- 
cal units,  must  be  multiplied,  to  obtain  the  numbers  expressing 
these  same  quantities  in  electromagnetic  or  electrostatic  units. 
For  example,  a  condenser  whose  capacity  is  4  microfarads  = 
4Xio-6  farads,  has  a  capacity  of  4Xio-J5  expressed  in  elec- 
tromagnetic   units    and    36X105    expressed    in    electrostatic 
units. 

Electrical  quantities  which  are  substituted  in  equations 
involving  centimeters,  grams,  and  seconds  must  be  in  absolute 
units,  and  the  results  of  such  equations  are  always  in  absolute 
units. 

LOGARITHMS. 

13.  Common  and  Natural  Logarithms. — If  z  is  the  common 
logarithm  of  y,  y  =  ioz.     If  z'  is  the  natural  logarithm  of  y, 
y  =  2.fjiS>z'.      The    base    of    natural    logarithms,    2.7 i8---,   is 
usually  designated  by  e.     We  shall  denote  the  common  loga- 
rithm by  log,  and  the  natural  logarithm  by  In.     Therefore, 
z  =log  y,  z'  =lrry.     Evidently  log  10  =  i,  In  2.718  =i. 

Natural  logarithms  are  obtained  as  the  result  of  mathe- 
matical operations,  such  as  integration.  Common  logarithms 
are  usually  employed  in  calculations  and  are  given  in  Table 
XLVIII. 

14.  Exponential  Equations. — If  the  exponent  is  a  fraction 
or  a  variable,  the  equation  may  be  simplified  by  taking  the 
logarithm  of  each  side.     Examples: 

(i).  y=az,  \ogy=z  log  a 
(2).  r  =  (i.y).3,  logr  =  .3  log  1.7  =.06912,  r  =  1.17 

15.  Conversion    of    Natural    Logarithms   to   Common  and 


10  INTRODUCTION. 

vice -versa. — If  we  equate  the  two  values  of  y  given  in  §13, 
ioz  =  2.7i82' 

:.z  =zr  log  2.718  =.4343  s'.     Or, 

logy  =  -4343  m:V  (8) 

Reciprocally,  z'  =  2.3032,  or, 

ln;y  =  2.303  logy  (9) 

Illustrations,  (i)  What  is  the  natural  logarithm  of  141 .5  ? 
We  find  in  common  logarithm  tables  that  log  141. 5  =2. 15076 

.'.  In  141.5=2.303X2.15076=4.953 

(2)  hry  =  1.217.  Find  y.  log  y  =  -4343  Xi.aiy  =.5285.  We 
find  in  a  table  of  common  logarithms  that  3.377  has  this 
logarithm,  and  is  therefore  the  value  of  y. 

ERRORS. 

16.  Inaccuracy  may   arise   from   several   different   causes: 

(1)  errors  of  observation,  due  to  the  inherent  limitations  of 
the  observer's  powers  of  observing,  judging  and  adjusting; 

(2)  instrumental  errors,  arising  from  imperfections  in  the  work 
of  the  instrument -maker  in  constructing  and  subdividing  the 
scale  used  by  the  observer;  (3)  mistakes,  such  as  the  mistaking 
of  an  8  for  a  3  on  a  scale ;  (4)  systematic  errors  due  to  faultiness 
in  the  general  method  employed.     The  first  may  be  decreased 
by  making  a  number  of  independent  observations  and  taking 
the  mean,  the  second  by  using  a  more  accurate  instrument, 
the  third  by  care  and  repetition,  the  fourth  by  seeking  a  better 
method. 

We  shall  consider  the  estimation  of  errors  of  the  first  type. 
If  we  have  a  large  number  of  observations  of  the  same  quantity 
we  may  reasonably  consider  the  result  reliable  within  the 
average  deviation  from  the  mean,  for  it  is  shown  in  treatises  upon 
probabilities  that  there  is  an  even  chance  of  the  observational 
error  being  greater  or  less  than 

.84  average  deviation 


\/number  of  observations. 


(10) 


ERRORS.  II 

If,  for  example,  we  are  measuring  the  pressure  of  a  gas,  and 
the  readings  of  the  manometer  height  for  six  independent  ad- 
justments are  31.33,  31.37,  3I-4<>,  31.31,  31.32,  and  31.38  cm., 
the  mean  is  30.35,  the  sum  of  the  deviations  (irrespective  of 
sign)  is  .20  and  the  average  deviation  is  .03.  We  therefore 
consider  that  the  manometer  height  is  31.35  and  that  this 
result  is  reliable  to  .03  cm.,  or,  /z  =31.35  ±  .03.  We  will  call 
.03  the  possible  error.  We  may  similarly  estimate  the  error  of 
a  result  involving  both  observation  and  calculation  if  we  have 
several  independent  determinations  of  this  result.  Usually, 
however,  we  introduce,  the  mean  observations  into  a  formula 
and  obtain  one  result,  and  we  will  now  consider  the  calculation 
of  the  possible  error  of  such  a  result. 

17.  If  the  final  result  is  a  sum  or  difference,  the  possible  error 
of  the  result  should  be  taken  as  the  sum  of  the  possible 
errors  of  the  two  terms,  for  one  may  be  too  great  and  the  other 
too  small. 

Illustration. — If  the  weight  of  an  exhausted  bulb  is  28.4325 
±.0003  gr.  and  the  weight  filled  with  a  vapor  is  29. 1007  ± 
.0003  gr.,  the  weight  of  the  vapor  is  .6682  ±.0006  gr.  Al- 
though the  possible  error  of  each  weighing  is  but  .001%,  the 
error  in  the  weight  of  the  vapor  is  .  i  % . 

1 8.  We   cannot   find   the   possible   error   of   a   product   or 
quotient  directly,  from  the  possible  errors  of  the  factors.     The 
percentage,   or  proportional,   error  of  each   factor  must  be 
determined  and  the  percentage  or  proportional  error  of  the 
result  is  their  sum. 

Let  the  volume  of  the  above  bulb  =982.3  ±.ic.c.  The 
density  is 

.6682^.0006        .  6682  (i+ .001) 


982.3=b-i          982.3  (i±  .0001) 
=  .00068024  (iifc.ooiqp.oooi)  (Equation  13) 
=  .  ooo68o±  .  oooooi 

The  main  purpose  of  an  estimation  of  the  possible  error  is  to 
determine  what  significant  figure  (note  §22)  should  be  retained 
and  therefore  the  order  (power  of  ten)  of  the  error  is  all  that  is 


12  INTRODUCTION. 

usually  required.  In  the  above  illustration,  .0001  is  negligible 
compared  with  .001.  .001  X.ooo68o24  is  .00000068  which 
we  call  .000001.  The  2  and  4  in  the  preliminary  calculation 
are  therefore  uncertain.  If  the  second  term  of  the  parenthesis 
of  the  next  to  the  last  equation  had  been  appreciable,  the  two 
proportional  errors  should  have  been  added  since  the  signs  are 
uncertain.  Instead  of  the  proportional  errors,  the  percentage 
errors  might  have  been  used. 

~'  I       =  .ooo68o±  .  i%=  .oooo68o±  .000001 


982.3+.  01% 

19.  The  possible  error  of  a  power  or  root  must  also  be  deter- 
mined from  the  percentage  or  proportional  error.  If  e  is  the 
error  in  a  quantity  A  , 

(/?     \  £> 

liH-j-J    (Equation    u),    where  —  r- 

is  the  proportional  error.  Therefore,  the  proportional  error 
of  the  square  of  a  quantity  is  twice  that  of  the  quantity 
itself,  of  the  square  root,  is  half  that  of  the  original  quantity. 
Illustration.  Suppose  the  above  bulb  is  a  sphere  and  we  wish 
to  determine  its  radius  r 


•4 


3X981-3  (i±.  =6.I6669  (I±.  00003)  = 

4^ 

6  .  1667+  .0002 


20.  Calculation  of  Possible  Error  of  Complicated  Expressions. 

—If  the  final  result  is  given  by  a  complicated  expression  in 
which  one  or  more  of  the  observed  quantities  appear  in  several 
factors,  the  errors  may  neutralize  each  other  to  some  extent; 
and  the  previous  directions  may  overestimate  the  error.  In 
such  cases  the  best  and  simplest  process  is  to  take  the  loga- 
rithm of  the  expression,  for  this  will  separate  the  factors,  and 
then  differentiate.  Since  the  differential  of  the  logarithm  of  a 
function  is  equal  to  the  differential  of  the  function  divided 
by  the  function,  the  differentials  of  the  logarithms  of  the 


ERRORS.  13 

factors  will  equal  their  proportional  errors,  if  we  let  the 
differentials  of  the  different  quantities  represent  their  possible 
errors. 

Illustration. — Derive  a  formula  for  the  proportional  error 
of  the  molecular  ref  ractivity  (§223). 

M  /*2-i 
~~ 


log  <7  =  log  Af  +  log  (/JL2  —  i)  -log  p— log 
Differentiating 

dG _d  M     2/j.dfi     dp     2judju 

G  M         JJL2  —  I          p  fJL  +  2 

fa  /Ltd  u.  dp 


M,  the  molecular  weight,  is  taken  from  Table  L,  and  is  as- 
sumed accurate,  or,  dM  =o.  Since  dp,  the  error  in  p,  may 
be  plus  or  minus,  the  sign  (minus)  is  chosen,  which  would 
give  the  greatest  possible  error.  The  signs  cannot,  however, 
be  disregarded  in  combining  the  second  and  fourth  terms,  since 
dfi  will  have  the  same  sign  in  both. 

Application.— Benzol.1     20°  (Yellow  (D)  light). 

M=y8,  ^=.8701^.003,  /*=  i  .5014+  .0003 
£=26.437 

dG_6Xi  .  5X  .0003      .003 
G==  (1.25)  (4.25)  "  .87 

=  . 0005  +  . 003  =  . 003  5  =  . 004 

If  the  proportional  error  in  G  is  .004,  the  actual  error  in  G 
is  .1  and  the  last  two  significant  figures  have  no  meaning;  we 
should  therefore  write 

(7  =  26. 4+ .  i 

1  The  organic  compound,  C6H6,  will  be  called  benzol,  throughout  this  book 
rather  than  benzene,  on  account  of  possible  confusion  with  benzine. 


14  INTRODUCTION. 

Limits  to  Calculations. 

21.  By  the  above  methods  the  possible  error  in  any  calcula- 
tion  from   experimental   quantities   may  be   deduced.     The 
magnitude  of  the  possible  error  in  any  calculation  indicates 
how  far  it  is  useful  and  desirable  to  carry  the  calculation. 
A  calculation  should  be  carried  as  far  as,  but  not  farther  than,  the 
first  doubtful  figure.     This  rule  must  be  applied  not  only  to  the 
calculation  of  the  final  result,  but  also  to  each  intermediate 
step.     When  a  calculation  is  carried  too  far,  useless  and  very 
unscientific  labor  is  expended,  and  when  it  is  not  carried  far 
enough  very  absurd  results  are  often  obtained. 

In  addition  and  subtraction  a  place  of  decimals  that  is 
doubtful  in  any  one  of  the  quantities  is  doubtful  in  the  result. 
In  multiplication  and  division  (performed  in  the  ordinary 
way)  decimal  places  that  have  not  been  determined  are  usually 
filled  up  by  zeros.  Any  figure  in  the  result  that  would  be 
altered  by  changing  one  of  these  zeros  to  5  is  doubtful. 

Abbreviated  Calculations. 

22.  Abbreviated  Multiplication. — In  the  long  multiplication 
and  the  long  division  of  numbers  obtained  by  experimental 
observations,   many  meaningless   figures   are   retained.     For 
example,  suppose  we  wish  to  determine  the  area  of  a  plate 
which  measures  21.64  cm.  by   17.49  cm.     Long  multiplica- 
tion gives  378.4836.     The  last  three  figures  are  absolutely 
uncertain  and  their  calculation  represents  wasted  time.     This 
may  be  avoided  by  the  use  of  logarithms  and  also  by  the 
approximate  method  of  multiplication  illustrated  below.     The 
first  figure  of  the  inverted  multiplier   (i)   is  placed  beneath 
the  last  figure  of  the  multiplicand  (4).     Each  figure  in  the 

multiplier  is  multiplied  into  the  figures  of  the  multipli- 
947I  cand,  above,  and  to  the  left,  carrying  from  the  first 
2164  number  above  and  to  the  right.  The  first  number  of 
I5J5  each  product  is  placed  in  the  first  column.  For  ex- 
IQ  ample,  7  times  6  is  42,  but  three  should  be  carried  from 
3785  7X4  (nearer  3  than  2).  Therefore  5  is  placed  in  the  first 


ERRORS.  15 

column.  7X1,  plus  4  carried,  is  n.  7X2,+!  carried,  is  15, 
etc.  The  pointing  off  may  be  done  by  inspection,  if  the  factors 
are  expressed  in  units  and  powers  of  10  (§2).  If  the  above 
numbers  had  been  216.4  and  .01749,  their  product  would  have 
been  2. 1 64X10*  XL 749  Xio-2  =3.785. 

If  the  number  of  significant  figures1  in  the  two  factors  is  not 
the  same,  add  a  cipher  to  the  less  and  replace  all  that  are  still 
in  excess  in  the  greater  by  zeros.  For  example,  2i7ooX 
.00867281  should  be  written  2.i7oXio4X8.673  Xio~3,  if 
the  ciphers  in  the  first  represent  undetermined  figures. 

23.  Abbreviated  Division. — If  we  knew  that  the  area  of  the 
plate  was  378.5  cm.  and  that  one  of  the  sides  was  21.64  and 
sought  the  length  of  the  other  side  by  long  division,  we  should 
obtain    17.4861   ....  cm.     The  abbreviated  division  illus- 
trated  below   obviates   the   calculation   of   the   meaningless 
figures.      The    first    division   and   the    first    subtraction   are 
similar  to  long  division.     One  figure  of  the  divisor  is  struck 
off  for  the  next  division,  leaving  216.     In  multiplying  by  the 
quotient,    7,  we  carry  from  the  figure  struck  2/03784(1749 
off;  that  is,  7  X6  is  42,  but  3  is  carried  from  2164 
7X4  giving  45,  etc.     The  divisor  for  the  next  I02° 
dividend  is,  similarly,  21,  which  goes  into  105 

4   times.       In   multiplying   out,  we   carry  as  86 

before.     4X1  =4.  and  2  is  carried  from  4X6,  J9 

etc.     The  final  dividend,    .19,  is  divided  by 
the  final  divisor,   2.      The  quotient  may  be  pointed  off  by 
inspection,  if  the  numbers  are  expressed  as  suggested  in  §2. 

Approximation  Formulae. 

24.  Products,   quotients,  or  powers,  of  expressions  of  the 
form  (i±e)  are  given  by  simple  formulae,  if  e  is  so  small  that  its 
square  and  higher  powers  may  be  neglected.     If  the  operations 
indicated  in  the  examples  below  are  carried  out  for  several 

1  The  significant  figures  are  the  figures  which  are  definitely  known,  counting 
from  the  right,  irrespective  of  the  decimal  point.  407.2  and  .004072  have  each 
four  significant  figures.  If  it  is  known  that  the  next  figure  is  o,  each  has  five 
significant  figures. 


1  6  INTRODUCTION. 

terms,  it  will  be  found  -that  the  approximate  expressions  are 
correct  within  these  limits. 

(n) 


;  \/  (i—e)  =i  ---  • 
2 


i  e 

—  ^  H  '    ' 


(\  i  — 
!-*)!  2 

(i±e2)  =  i±eI±e2  '  (12) 


d3) 


- 

Therefore,  the  geometric  mean  of  two  nearly  identical 
numbers  is  equal  to  the  arithmetic  mean,  within  the  above 
limits. 

Illustrations.  —  (i)    If  t  is  small,    xl  i  H  --  =  iH  --  -; 


80      i+~ 


—  i--oi4=-986: 


APPARATUS. 

The  more  common  apparatus  of  physical  chemistry  is 
described  in  the  following  pages.  Apparatus  which  is  only 
used  in  one  or  two  experiments  is  described  under  the  respect- 
ive experiments,  but  a  reference  to  such  descriptions  will  be 
found  in  this  Introduction. 


APPARATUS.  17 

25.  The  Use  of  a  Vernier. — The  vernier  is  a  contrivance  for 
reading  to  fractions  of  the  unit  in  which  a  scale  is  graduated. 
It  is  a  second  scale  parallel  to  the  main  scale  of  the  instrument 
and  so  divided  that  n  of  its  units  equal  n  —  i  of  the  units  of  the 
scale.  If  5  is  the  length  of  a  scale  unit  and  v  that  of  a  vernier 
unit, 

nv=(n  —  i)  s  .'.  s—  v  =  s  +  n 

Or  the  unit  of  the  vernier  is  less  than  that  of  the  scale  by 
one-wth  of  a  scale  unit. 

If  we  did  not  have  a  vernier  there  would  be  something  in 
the  nature  of  an  index  to  indicate  what  division  of  the  scale 
should  be  read  in  making  a  certain  measurement,  and  fractions 


I  I 


LL  J  L 


FIG.  i.  FIG.  2. 

would  be  estimated  by  eye.  The  zero  of  the  vernier  is  taken 
as  an  index,  the  whole  number  of  scale  divisions  being  the 
number  just  below  the  zero  of  the  vernier,  while  the  fraction 
of  a  scale  division  is  determined  with  the  vernier.  If  the  with 
division  of  the  vernier  coincides  with  the  scale  division,  the 
zero  of  the  vernier  must  be  m— wths  of  a  scale  unit  from  the 
scale  division  just  below  it.  The  reading  of  the  scale  of 
Fig.  i,  would  be  5.235,  for  the  coincidence  is  evidently  between 
the  third  and  fourth  vernier  divisions.  The  most  common 
verniers  with  linear  scales  are  those  reading  to  tenths,  twen- 
tieths or  twenty-fifths,  and  with  circular  scales,  the  usual 
vernier  reads  to  thirtieths. 

26.  Vernier  Caliper. — The  ^vernier  caliper  consists  of  a 
straight  graduated  bar  and  two  jaws  at  right  angles  to  it, 
one  of  which  is  fixed,  while  the  other  is  movable.  The  position 
of  the  movable  jaw  can  be  accurately  determined  by  means  of 


r8 


INTRODUCTION. 


the  scale  and  a  vernier  which  should  read  zero  when  the  jaws 
are  in  contact.  (If  this  be  not  the  case,  allowance  must  be 
made  for  the  zero  reading.) 

27.  Micrometer  Microscope. — The  micrometer  microscope  is 
a  microscope  with  cross  hairs  at  the  focus.  In  one  type  of 
instrument  these  cross  hairs  are  movable  by  a  micrometer 
screw.  In  the  other  and  more  common  type  the  whole  micro- 
scope is  moved  by  a  micrometer  screw  (Fig.  3 ) .  The  best  instru- 
ments have  both  adjustments.  The  rotations  of  the  screw  are 


T 


FIG.  3. 

read  on  a  fixed  linear  scale,  the  fraction  of  a  rotation  is  read 
by  a  circular  scale  attached  to  the  screw,  and  thus  the  amount 
of  movement  is  ascertained  if  the  pitch  of  the  screw  is  known . 
The  pitch  is  best  determined  by  measuring  a  definite  length 
on  a  reliable  scale,  placed  in  the  field  of  view. 

28.  Comparator. — The  comparator  consists  essentially  of  a 
pair  of  microscopes  movable  along  a  horizontal  bar  to  which 
they  are  at  right  angles.  The  length  to  be  measured  is  placed 
under  the  microscopes.  The  eye-piece  of  each  microscope 
is  first  focused  clearly  on  the  cross  hairs  and  then  the  whole 
microscope  is  focused  without  parallax  on  the  point  to  be 
observed,  and  the  position  is  adjusted  until  the  image  of  the 
point  coincides  with  the  intersection  of  the  cross  hairs.  The 
object  is  then  removed  and  a  good  scale  put  in  its  place,  and 
a  reading  of  the  scale  gives  the  required  length,  this  reading 
being  facilitated  by  the  use  of  the  micrometer  screws. 


APPARATUS.  19 

29.  Cathetometer. — The  cathetometer  is  a  vertical  pillar, 
supported  on  a  tripod  and  leveling  screws,  and  capable  of 
rotation  about  its  axis ;  the  pillar  is  graduated  and  a  horizontal 
telescope  with  cross  hairs  is  borne  by  a  carriage  that  travels 
on  the  pillar  and  can  be  clamped  at  any  desired  position.  A 
slow-motion  screw  serves  for  accurate  adjustment  of  the  posi- 
tion of  the  telescope. 

Adjustments. — (i)  The  intersection  of  the  cross  hairs  must 
be  in  the  optical  axis  of  the  telescope.  To  secure  this,  focus 
the  intersection  of  the  cross  hairs  on  some  mark,  rotate 
the  telescope  about  its  own  axis,  and  see  whether  the  cross 
hairs  remain  on  the  mark.  If  not,  the  adjusting  screws 
of  the  cross  hairs  must  be  changed  until  this  is  attained. 

(2)  The  level  must  be  properly  adjusted. — Level  the  telescope 
until   the  bubble  comes  to  the  center  of  the  scale.     Turn 
the  level  end  for  end.     If  the  bubble  does  not  come  to  the 
same  position,  the  level  must  be  adjusted  until  it  will  stand 
this  test. 

(3)  The  scale  must  be  vertical. — If  there  are  separate  levels 
for  the  shaft,  this  is  readily  attained.     If  there  is  but  one 
level  for  telescope  and  shaft,   this  and  the  next  adjustment 
must  be  made  simultaneously. 

(4)  The  telescope  must  be  perpendicular  to  the  scale. — The 
top  of  the  scale,  T,  may  be  regarded  as  having  two  degrees  of 
freedom — first,   parallel   to   the   line   of  two  leveling  screws 
of  the  base,  A  and  B;  second,  in  a  line  through  the  third 
leveling  screw,  C,  perpendicular  to  AB.     If  A  and  B  be  screwed 
equal  amounts  in  opposite  directions,  T  will  move  parallel  to 
AB.     If  C  only  be  turned,  T  will  move  perpendicular  to  AB. 

First,  make  the  telescope  horizontal  and  parallel  to  AB. 
Turn  the  shaft  through  180°.  It  is  evident  that  if  the  tele- 
scope makes  an  angle  (/>  with  the  normal  to  the  scale,  turning 
the  scale  through  180°  will  cause  the  telescope  to  make  an 
angle  2<£  with  its  former  direction.  Hence,  with  the  leveling 
screw  of  the  telescope,  correct  half  the  error  in  the  level, 
and,  by  turning  A  and  B  equally  in  opposite  directions,  correct 
the  remainder.  Turn  the  telescope  to  the  first  position  and 


20  INTRODUCTION. 

repeat  the  above  adjustments,  then  to  the  second,  and  repeat 
as  many  times  as  is  necessary.  Then  turn  the  telescope  normal 
to  AB  and  adjust  by  C.  When  the  adjustment  is  complete, 
turning  the  shaft  through  360°  will  not  alter  the  position  of 
the  bubble. 

Adjustments  (i)  and  (2)  are  not  usually  required.  The 
eye-piece  of  the  telescope  is  focused  until  the  cross  hairs 
are  seen  clearly,  and  the  focus  of  the  objective  is  changed 
until  the  object  is  seen  very  distinctly  and  without  parallax, 
i.e.,  with  no  relative  motion  with  respect  to  the  cross  hairs 
when  the  eye  is  moved  about. 

THE  BALANCE. 

30.  Precautions  in  Use  of  Balance. 

1.  Note  the  maximum  load  that  may  be  placed  on  the 
balance  and  take  care  not  to  exceed  it. 

2.  Always  stop  the  swinging  of  the  beam  by  means  of  the 
arrestment,  before  in  any  way  altering  the  load  on  the  pans. 

3.  Do  not  stop  the  swinging  of  the  balance  with  a  jerk. 
It  is  best  to  stop  it  when  the  pointer  is  vertical. 

4.  To  set  the  beam  in  vibration,  do  not  touch  it  with  the 
hand,  but  raise  and  lower  the  arrestment. 

5.  Place  the  large  weights  in  the  center  of  the  pan. 

6.  Make  final  weighings  with  the  case  closed. 

7.  Do  not  place  anything  in  contact  with  a  pan  that  is 
liable  to  injure  it. 

8.  Avoid,  if  possible,  weighing  a  hot  body. 

9.  Never  handle  the  weights  with  the  fingers,  as  this  may 
change  some  of  the  weights  appreciably.     Always  use  the 
pincers. 

Instead  of  using  milligram  weights,  it  is  customary  to  use 
a  rider  of  .01  g.,  which  can  be  placed  on  the  beam  at  various 
distances  from  the  center.  The  beam  is  for  this  purpose 
graduated  into  10  divisions,  which  may  be  still  further  sub- 
divided. Thus  the  .010  g.  rider  placed  at  the  division  of  4  of 
the  beam  is  equivalent  to  .004  g.  placed  on  the  pan. 


THE    BALANCE.  21 

3 1 .  Weighing  by  Oscillations. — The  zero-point  of  the  balance 
is  the  position  on  the  scale  behind  the  pointer  at  which,  the 
pans  being  empty,  the  pointer  would  ultimately  come  to  rest; 
it  must  not  be  confused  with  the  zero  of  the  scale.  As  much 
time  would  be  wasted  in  always  waiting  for  the  pointer  to 
come  to  rest,  the  zero  of  the  balance  is  best  obtained  from  the 
swings  of  the  pointer.  For  this  purpose,  readings  of  the 
successive  "turning-points"  are  made  as  follows — three 
successive  "turning  points"  on  the  right  and  the  two  inter- 
mediate ones  on  the  left,  or  vice  versa;  e.g., 

Turning  points. 
L.  R. 

+  2.1 
+  2.0 


Mean,  —1.13  +2.05 

-1-13 

Zero  point  =  +  0.92-1-2= +0.46 

By  taking  an  odd  number  of  successive  turning-points  on  one 
side  and  the  intermediate  even  number  on  the  other  side  and 
then  averaging  each  set,  we  eliminate  the  effect  of  the  gradual 
decrease  of  amplitude  of  the  swing. 

The  resting-point  of  the  balance  with  any  loads  on  the 
pans  is  the  point  at  which  the  pointer  would  ultimately  come 
to  rest,  and  is  found  in  the  same  way  as  the  zero-point.  If 
the  resting-point  should  happen  to  be  the  same  as  the  zero-point, 
the  weight  of  the  body  on  one  pan  is  immediately  found  by 
the  weights  on  the  other  pan  and  the  position  of  the  rider. 
Usually,  however,  this  will  not  be  so.  It  will  then  be  neces- 
sary to  find  the  nearest  resting-point  to  the  right  of  the  zero, 
and  then,  after  altering  the  rider  one  place,  to  find  the  nearest 
resting-point  on  the  other  side  of  the  zero.  By  interpolation, 
the  change  of  the  position  of  the  rider  necessary  to  make 
the  resting-point  coincide  with  the  zero-point  is  deduced. 
For  example,  the  zero  is +0.46  (Fig.  4);  with  the  rider  at  4 
the  resting-point  is  +0.51;  with  the  rider  at  5  the  resting- 


22  INTRODUCTION. 

point  is  H-o.io.  By  changing  the  rider  from  4  to  5,  o.ooi  g. 
was  added.  To  bring  the  resting-point  to  the  zero  we  should 
have  added  .05  -f-(o.5i  —  o.io)  of  .001  g.,  or  .0001  g.  approx- 
imately. Hence  the  weight  of  the 
—  body  is  the  weight  on  the  pan  plus 

0.0041  g. 

FIG.  4.  32'   The  Arms  of  the  Balance  May  be 

Unequal.      If  this  be   so,  the  weight 

obtained  above  will  not  be  the  true  weight.  To  eliminate  this 
error  the  body  must  be  changed  to  the  other  pan  and  another 
weighing  made.  If  /  be  the  length  of  the  left  arm  and  r  that 
of  the  right  and  if  u  be  the  counterbalancing  weight  when 
the  body  is  in  the  left  pan  and  v  when  it  is  in  the  right, 
while  w  is  the  true  weight  of  the  body,  then 

lw  =  ru,  lv  =  rw, 
.-.  w  =  \/uv  =  —  (u  +  v).     (Eq.-i4.)  (15) 

33.  The  buoyancy  of  the  air  on  the  weights  and  on  the  body 
must  be  allowed  for  in  accurate  work.  To  the  apparent  weight 
of  the  body  must  be  added  a  correction  equal  to  the  weight  of 
the  air  displaced  by  the  body  and  from  the  apparent  weight 
must  be  subtracted  the  weight  of  the  air  displaced  by  the 
weights.  Let  m  =  apparent  mass  of  body ;  p  =  density  of  body ; 
d=  density  of  air.  The  density  of  ordinary  brass  weights  is 
8.4.  Since 

m 
P 

is  the  approximate  volume,  the  correction  to  be  added  for 
buoyancy  on  the  body  is 

^-d, 
P 

and  the  correction  to  be  subtracted  for  the  buoyancy  on  the 
weights  is 

m 


THE    BALANCE. 


or,  the  total  correction  is 


md  (  —  - 
P 


(16) 


The  density  of  dry  air  at  o°  and  76  cm.  pressure  is  .001293. 
The  density  at  any  other  pressure  and  temperature  can  be 
calculated  by  Equation  53.  Unless  extreme  accuracy  is  re- 
quired, the  density  under  ordinary  working  conditions  may  be 
assumed  as  .001 2 .  Table  IV  gives  the  values  of  this  correction 
in  milligrams,  or, 


IOOOX  .0012 


for  different  values  of  p.  An  approximate  value  of  p,  the 
density  of  a  body,  may  be  obtained  from  the  uncorrected 
weight  and  volume.  For  example,  the  uncorrected  weight 
of  a  body  was  6.2341  gr.  and  the  approximate  density  was  2.5, 
the  approximate  volume  was  therefore  2.5  and  the  correction 
was  6.2X-34=2.i  milligrams.  Therefore  the  weight  reduced 
to  vacuum  was  6.2362  grams. 

TABLE  IV. 
Reduction  of  Weighings  to  Vacuum. 

(The  density  of  air  is  assumed  as  .0012  and  that  of  the  weights 
as  8.4.) 

Density  of 
Body 

•5 
.6 


34.  Correction  of  a  Set  of  Weights. — Weights  by  good  makers 
are  usually  so  accurate  that  errors  in  them  may  for  most 
purposes  be  neglected.  But  when  less  perfect  weights  are 


erection 

Density  of 

Correction 

illigrams) 

Body 

(milligrams) 

2  .26 

2-5 

•34 

.86 

3- 

.26 

•57 

3-5 

.  20 

•36 

4- 

.16 

.19 

5- 

.10 

.06 

6. 

.06 

.86 

8. 

.01 

•7i 

10  . 

.02 

.61 

15- 

—  .06 

•52 

20. 

—  .08 

•46 

24  INTRODUCTION. 

to  be  used  or  when  weighings  are  to  be  made  with  the  highest 
possible  degree  of  accuracy,  the  errors  in  the  weights  must 
be  carefully  ascertained. 

We  shall  suppose  that  a  100  g.  box  of  weights  is  to  be 
tested  and  that  a  reliable  100  g.  weight  is  supplied  as  a  stand- 
ard and  that  an  accurate  10  mg.  rider  is  supplied  for  making 
the  weighings.  The  weights  of  the  box  will  be  denoted  by 
ioc/,  50',  20',  20",  10',  and  so  on,  and  the  sum  5'  +  2'  +  2"  +  i' 
by  10".  To  find  the  six  unknown  quantities,  100',  50',  20', 
20",  ic/,  10",  we  'must  make  six  weighings  and  obtain  six 
relations  between  these  quantities.  Such  a  set  of  weighings 
are  indicated  in  the  following  table.  Each  should  be  performed 
by  the  method  of  double-weighing  just  described, 

ic/  =  10"  -fa 

20'  =io'  +  io"+6 


100'  =  50'  +  20'  +  20"  +  10'  +c 

IOO    =IOC/+/ 

To  solve  these  equations,  substitute  the  value  of  10'  given 
by  the  first  in  the  second;  then  substitute  the  value  of  20' 
given  by  the  second  in  the  third,  and  so  on  to  the  last,  when 
the  value  of  10"  in  terms  of  the  standard  100  and  a,  6,  c, 
d,  e,  f  will  be  obtained.  The  calculation  of  the  other  quan- 
tities will  then  present  no  difficulty.  To  standardize  the 
box  completely  the  same  process  must  be  applied  to  10',$',  2' 
2"  ',  i',  i"  and  sirnilarly  to  the  smaller  weights. 

DENSITY  OF  LIQUIDS. 

35.  Pyknometer.  —  A  simple,  accurate  method  of  finding  the 
density  of  a  liquid  is  to  weigh  a  suitable  vessel,  (a)  empty, 
(6)  filled  with  water  at  a  known  temperature,  and  (c)  filled  with 
the  liquid.  The  first  two  weighings,  together  with  the  density 
of  water  (Table  LI),  give  the  volume  of  the  vessel  and  the 
first  and  last  weighings  give  the  mass  of  the  liquid.  The 
quotient  of  the  mass  divided  by  the  volume  is  the  density. 


DENSITY    OF    LIQUIDS.  25 

The  most  convenient  form  of  vessel  is  the  Ostwald  pyknometer 
(Fig.  5).  The  vessel  is  properly  filled  when  the  liquid  fills  it 
completely  from  the  mark  B  to  the  tip  of  the  capillary  A . 

An  Ostwald  pkynometer  is  very  convenient  as  a  weighing 
pipette.  It  is  partly  filled  with  the  liquid  and  weighed. 
The  desired  amount  of  liquid  is  expelled  from  A  by  blowing  at 
B,  and  the  exact  amount  of  liquid  expelled  is  determined  by 
reweighing. 


FIG.  5. 

36.  Mohr-Westphal  Specific  Gravity  Balance. — This  is 
a  convenient  form  of  hydrostatic  balance  for  finding  the 
density  of  a  liquid  by  determining  the  buoyancy  of  the  liquid 
on  a  float  hung  from  an  arm  of  the  balance  and  immersed 
in  the  liquid.  Instead  of  weights,  riders  are  used,  the  arm  of 
the  balance  from  which  the  float  hangs  being  graduated  into 
ten  divisions.  The  float  is  made  of  such  a  size  that  when 
hanging  in  air  from  the  graduated  arm  of  the  balance  (which 
is  less  massive  than  the  other  arm)  it  will  just  produce  equilib- 
rium. Four  riders  of  different  mass  are  employed,  each  one 
being  ten  times  as  heavy  as  the  next  smaller.  The  largest 
rider  is  of  such  a  size  that  if  the  float  hanging  from  the  balance 
be  immersed  in  water  at  15°  C.,  the  addition  of  the  rider  to  the 
hook  at  the  end  of  the  beam  will  restore  equilibrium.  Hence 
it  counterbalances  the  buoyancy  of  the  water  on  the  float 
Thus  it  is  evident  that  if  the  water  be  replaced  by  a  liquid  of 
unknown  density  at  the  same  temperature  (so  that  the  volume 
of  the  float  is  the  same)  and  if  the  largest  rider  under  the  cir- 


26  INTRODUCTION. 

cumstances  produces  equilibrium  when  placed  at  the  sixth 
division,  then,  for  equal  volumes,  this  liquid  can  weigh  only 
six-tenths  as  much  as 'water,  or  its  density  is  0.6.  A  second 
rider,  one-tenth  as  heavy  as  the  first,  would  evidently  enable 
us  to  carry  the  process  one  decimal  place 
farther,  etc.  For  liquids  of  a  density  exceed- 
ing unity,  another  rider  equal  to  the  largest 
must  be  hung  from  the  ends  of  the  beam, 
FIG.  6.  and  stiH  a  third  may  be  necessary  for  liquids 
of  density  above  2. 

From  the  above  it  will  be  seen  that  (i)  the  balance  must 
be  adjusted  by  the  leveling  screw  on  the  base  until  the  end 
of  the  beam  is  opposite  the  stud  in  the  framework  when  the 
float  is  suspended  in  the  air;  (2)  the  beaker  must  always 
be  filled  to  the  same  level,  that  level  being  such  that  when 
the  liquid  is  water  at  15°  C.  the  balance  is  in  equilibrium 
with  the  largest  rider  hanging  above  the  float,  and  (3)  the 
liquid  tested  should  be  at  15°.  If,  however,  the  expansion 
of  the  glass  sinker  is  calculated  from  the  coefficient  of  cubical 
expansion  (Table  LV) ,  it  will  be  found  that  the  error  in  using 
the  balance  at  any  temperature  between  10°  and  20°  is  within 
the  errors  of  observation. 

The  density  of  a  liquid  may  be  approximately  determined 
with  a  carefully  graduated  hydrometer  of  variable  immersion. 

For  measurement  of  viscosity,  see  experiment  XI,  and  for 
measurement  of  surface  tension  see  experiment  X. 

MEASUREMENT  OF  PRESSURE. 

37.  The  open-tube  manometer  consists  of  a  U-tube  containing 
mercury,  or,  some  lighter  liquid,  such  as  castor  oil,  if  the 
change  of  pressure  is  small.  One  side  is  connected  to  the 
vessel  in  which  it  is  desired  to  determine  the  pressure  and 
the  other  side  is  open  to  the  air  (see  Fig.  57).  The  pressure 
in  the  vessel  is  the  corrected  barometric  pressure  plus  or  minus 
the  difference  in  level  in  the  two  arms,  reduced  to  mercury 
at  zero  degrees.  If,  for  example,  the  mercury  in  the  arm 


BAROMETER.  2J 

next  the  vessel  is  12.05  cm-  below  the  level  in  the  open  arm, 
and  the  temperature  is  20°  and  the  corrected  barometer  height 
is  74.11  cm.,  the  pressure  in  the  vessel  is  74.11  +12.05 
(i —20  X- 000181)  =86.07  cm-  [-000181  is  the  coefficient  of 
cubical  expansion  of  mercury  (Table  LIII).] 

38.  An  open-tube  manometer  is  obviously  limited  to  moderate 
pressures.     For    higher    pressures    a    closed-tube    manometer 
must  be  used.     This  is  similar  to  an  open  manometer  except 
that  a  closed  tube  is  attached  to  what  would  be  the  open  arm. 
The  pressure  in  the  vessel  attached  to  the  other  end  is  equal  to 
the    corrected    difference   of  level   plus  the   pressure   of  the 
inclosed  air.     The  latter  is  calculated  from  the  volume  of  the 
inclosed  air  and  the  initial  volume  and  pressure  (Equation  38'). 

BAROMETER. 

39.  Cistern  Barometer. — The   cistern  is  raised  or  lowered 
by  means  of  the  screw  at  the  bottom  until  the  mercury  in  the 
cistern  just  meets  an  ivory  stud  near  the  side  of  the  cistern. 
The  zero  of  the  scale  is  the  tip  of  this  stud.     A  collar,  to  which 
is  attached  a  vernier,  is  so  placed  that  the  top  of  the  meniscus 
of  the  mercury  column  is  tangent  to  the  plane  of  the  two 
lower  edges.     The  height  of  the  barometer  should  be  reduced 
to  zero  by  the  formula 

h0  =  h  (i  —  .000162  t)  (17) 

where  h  is  the  observed  height,  h0  the  height  at  o°  and  t  the 
temperature  centigrade.  For,  the  expansion  of  the  mercury 
will  increase  the  height  in  the  ratio  (i  —.000181  t),  and  the 
expansion  of  the  brass  scale  will  reduce  the  apparent  height 
in  the  ratio  (i  —.000019  0- 

40.  Syphon  Barometer. — By  means  of  two  scales,  graduated 
in  opposite  directions  from  a  common  zero,  the  position  of  the 
mercury  is  read  in  both  arms.     The  length  of  the  mercury 
column  is  the  sum  of  the  two  readings.     The  scale  is  usually 
etched  directly  on  the  glass,  and  since  the  glass  expands  much 
less  than  brass,  the  correction  is  a  little  greater.     The  correc- 
tion formula  is 

h0  =  h  (i  -  .000173)  (18) 


28 


INTRODUCTION. 


Since  the  mercury  may  adhere  to  glass  to  some  extent, 
barometer  tubes  should  be  tapped  before  reading.  Table  V  gives 
the  barometric  correction  for  different  pressures  and  tem- 
peratures. The  correction  for  intermediate  pressures  must  be 
found  by  interpolation  or  direct  calculation  by  the  above 
equations. 

41.  If  the  pressure  is  not  expressed  in  absolute  measure 
(§5),  the  height,  reduced  to  o°,  should  further  be  reduced  to 
the  height  under  the  standard  value  of  gravity,  980.6.  If 
h0  is  the  corrected  height  where  the  acceleration  of  gravity  is  g, 
the  final  corrected  height  is 


(19) 


TABLE  V. 
Reduction  of  Barometer  Readings  to  o°. 

(The  corrections  below  are  in  mm.  and  are  to  be  substracted.     The 
uncorrected  height  is  in  cm.) 


Temp. 

Brass  Scale 

Glass  Scale 

72 

73 

74 

75 

76 

77 

78 

74 

75 

76 

77 

78 

i5 

r-75 

1.77 

1.81 

1-83 

i  86 

1.88 

1.91 

1.92 

1.94 

1.97 

2.IO 

2.OO 

2.02 

16 

1.87 

1.89 

i-93 

1.96 

1.98 

2.01 

2.03 

2.05 

2.07 

2.132.16 

J7 

1.98 

2.10 

2.OI 

2.13 

2.0_5 

2.17 

2.08 

2.20 

2.10 

2.13   2.l6 

i 
2.17 

2.30 
2.43 

2.  2O 

2.23 
2.36 
2.49 
2.62 
2.76 

2.26 
2-39 

2    29 
2-43 

18 

2.23 

2.26 

2.29 

2-33 
2.46 

2-59 

!Q 

2.22 

2.25 

2.29 

2.32 

2-35 

2.38  2.41 

2-53 

2.56 
2.69 
2.83 

20 

2-33 

2-37 

2.41 

2.44 

2.56 

2.47 

2-51 

2-54 

'2.56 

2.66 

2.79 

21 

2.45 

2.48 

2-53 

2.60 

2.72 

2,3 

2.67 
2.79 

2.68 
2.81 

2.72 

«             22 

2-57 

2.60 

2.65 

2.69 

2.81 

2.93 

2.76 

2.85 

2.89  2.92 

2.96 

23 

2.6812.72 

2-77 
2.89 

2.84 

2.97 
3-09 

2  88 

2.92 
3-°5 
3-17 

2.94 

2.98  3.02 
3-11  3-J5 

3.06 

3-10 

24 

2.80 

2.84 

3.01 
3-J3 

3.06 

3-J9 

3-23 
336 

25 

2.92  2.96 

3-01 

3.05 

3-!9 

3-23 

3.28 

3-32 

HYGROMETRY.  29 

If  the  value  of  g  is  not  known  at  a  place  in  latitude  (j>  and 
d  meters  above  the  sea  level,  it  may  be  calculated  by  Broch's 
formula. 

£  =  980.6  (i  —  .0026  cos  2 </>—  . 000000196 d)          (20) 

HYGROMETRY. 

Three  methods  may  be  used  for  studying  the  hygrometric 
state  of  the  atmosphere.  The  first  method  (A)  determines  the 
dew  point,  the  second  (B)  determines,  indirectly,  the  actual 
vapor  pressure,  and  the  third  (C)  determines  the  relative 
humidity. 

42.  (A)  Regnault's  Hygrometer. — A  thin  silvered  glass  test- 
tube  is  half  filled  with  ether.     The  test-tube  is  tightly  closed 
by  a   cork   through  which   passes   a   sensitive   thermometer 
which  gives  the  temperature  of  the  ether.     Two  glass  tubes 
also  pass  through  the  cork,  one  extending  to  the  bottom,  the 
other  ending  below  the  cork.     An  aspirator  gently  draws  air 
from  the  shorter  tube.     The  ether  is  evaporated  by  the  air 
bubbles  and  the  entire  vessel  cools.     The  silvered  surface  and 
the  thermometer  are  watched  through  a  telescope  and  the  tem- 
perature is  read  the  moment  moisture  appears  on  the  metal. 
The  air  current  is  stopped  and  the  temperature  of  disappearance 
of  the  moisture  is  observed.     This  is  repeated  several  times  and 
the  mean  is  taken  as  the  dew  point.     (§114).     The  detection  of 
moisture  is  facilitated  by  observing  at  the  same  time  a  similar 
piece  of  silvered  glass  which  covers  a  part  of  the  test-tube,  but 
which  is  insulated  from  it.     The  temperature  of  the  air  should 
also  be  carefully  determined,  preferably  with  a  thermometer 
in  a  similar  apparatus  where  there  is  no  evaporation. 

43.  (B)  Wet  and  Dry  Bulb  Hygrometer. — Two  thermometers 
are  mounted  a  few  inches  apart.     About  the  bulb  of  one  is 
wrapped  muslin  cloth  to  which  is  attached  a  muslin  wick 
dipping  in  water.     The  other  is  bare.     The  temperatures  of 
both  are  read  when  they  have  become  steady.     The  tempera- 
ture of  the  first  thermometer  will  be  lower  than  that  of  the 
bare  thermometer,  on  account  of  the  evaporation  of  the  water. 
From  the  difference  of  temperature  of  the  two  thermometers 


INTRODUCTION. 


and  the  temperature  of  the  bare  thermometer,  the  actual 
vapor  pressure  may  be  determined  with  the  aid  of  empirical 
tables  (see  Table  VI).  The  accuracy  will  be  much  increased' 
if  the  "wet"  thermometer  is  in  motion,  e.g.,  attached  to  a 
pendulum. 

TABLE  VI. 
Wet  and  Dry  Bulb  Hygrometer. 

(Actual  vapor  pressures  (mm.)  for  different  temperatures  of  dry 
thermometer  and  various  differences  of  temperature  between  the  two 
thermometers. 

The  first  vertical  column  gives  the  temperature  of  the  dry-bulb 
thermometer.  The  first  horizontal  line  gives  the  difference  between 
the  two  thermometers.  Since  the  difference  is  zero  if  the  air  is  satu- 
rated, the  second  vertical  column  gives  the  saturated  vapor  pressure 
for  the  corresponding  temperatures  in  the  first  column.) 


t°C. 

O 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

I  I 

o 

4-6 

3-7 

2.9 

2.1 

J-3 

i 

4-9 

4.0 

3-2 

2.4 

1.6 

0.8 

2 

5-3 

4.4 

3-4 

2-7 

1.9 

I.O 

3 

5-7 

4-7 

3-7 

2.8 

2.2 

i-3 

4 

6.1 

5-1 

4.1 

3-2 

2.4 

1.6 

0.8 

5 

6-5 

5-5 

4-5 

3-5 

2.6 

1.8 

I.O 

6 

7.0 

5-9 

4.9 

3-9 

2-9 

2.0 

i.i 

7 

7-5 

6.4 

5-3 

4-3 

3-3 

2-3 

1.4 

0.4 

8 

8.0 

6.9 

5-8 

4-7 

3-7 

2.7 

i-7 

0.8 

9 

8.6 

7-4 

6-3 

5-2 

4.1 

3-1 

2.1 

i.i 

0.2 

10 

9.2 

8.0 

6.8 

5-7 

4-6 

3-5 

2-5 

»-5 

o-5 

1  1 

9.8 

8.6 

7-4 

6.2 

5-1 

4.0 

2-9 

1.9 

09 

12 

IO-S 

9.2 

8.0 

6.8 

5-6 

4-5 

3-4 

2-3 

!-3 

J3 

I  1.2 

9.8 

8.6 

7-3 

6.2 

5-o 

3-9 

2.8 

I-7 

14 

II.9 

10.6 

9.2 

8.0 

6.7 

56 

4-4 

3-3 

2.2 

i.i 

15 

12.7 

n-3 

9-9 

8.6 

7-4 

6.1 

5-° 

3-8 

2-7 

1.6 

°-5 

16 

I3-S 

12.  I 

10.7 

9-3 

8.0 

6.8 

5-5 

4-3 

3-2 

2.1 

I.O 

!? 

14.4 

I3.0 

n-5 

10.  1 

8.7 

7-4 

6.2 

4.9 

3-7 

2.6 

i-5 

0.4 

18 

i'5-4 

13-8 

12.3 

10.9 

9-5 

8.1 

6.8 

5-5 

4-3 

3-i 

2.O 

0.9 

19 

16.4 

14-7 

13.2 

11.7 

10.3 

8.9 

7-5 

6.2 

4-9 

3-7 

2-5 

1.4 

20 

17.4 

IS-7 

14.1 

12.6 

i  i.i 

9-7 

8-3 

6.9 

5-6 

4-3 

3-1 

1.9 

2  I 

18-5 

16.8 

*5-x 

I3-S 

12.0 

10.5 

9.0 

7.6 

6-3 

5-o 

37 

2-5 

22 

19.7 

17.9 

1-6.2 

14-5 

I2.9 

11.4 

9-9 

8.4 

7.0 

5-7 

4.4 

3-1 

23 

20.9 

19.0 

J7-3 

15.6 

13-9 

12.3 

10.8 

9-2 

7-8 

6.4 

5-1 

3-8 

24 

22.2 

20.3 

18.4 

16.6 

14.9 

J3-3 

11.7 

IO.I 

8-7 

7-2 

5-8 

4-5 

25 

23.6 

21.6 

19.7 

17.8 

16.0 

J4-3 

12.7 

I  I.I 

9-5 

8.0 

6.6 

5-2 

26 

25.0 

22.9 

21.  0 

19.0 

17.2 

15-4 

*3-7 

12.  I 

IO-5 

8.9 

7-4 

6.0 

27 

26.5 

24.9 

22-3 

20.3 

18.4 

16.6 

14.8 

I3-1 

11.4 

9-8 

8-3 

6.8 

28 

28.1 

25-9 

23-7 

21.7 

19.7 

!7.6 

16.0 

14.2 

12.5 

10.8 

9-2 

7-7 

29 

29.8 

27-5 

25-3 

23.1 

2  I.I 

19.1 

17.2 

15-3 

13.6 

n.9 

IO.2 

8.6 

3° 

31.6 

29.2 

26.9 

24.6 

22-5 

20.5 

18.5 

16.6 

14.7 

13.0 

I  1.2 

9.6 

MEASUREMENT    OF    TEMPERATURE.  31 

Illustration. — The  temperature  of  the  wet  thermometer 
was  21.4°  when  the  dry  thermometer  read  26°.  Finding  26° 
'in  the  first  column,  and  then  going  along  the  horizontal  line 
we  find  1 6. i,  approximately,  for  the  pressure  corresponding 
to  the  difference  of  4.6°.  The  saturated  vapor  pressure  for 
26°  is  given  in  the  second  column  (25).  The  relative  humidity 
is  therefore 

16.1 

—  =  .642 

and  the  dew  point  is  the  temperature  in  the   first   column 
corresponding  to  16.1  in  the  second  column,  or  18.7°. 

44.  (C)  Chemical  Hygrometer. — Fill  three  ordinary  balance 
drying  vessels  with  pumice.     Saturate  two  with  strong  sul- 
phuric acid  and  the  third  with  distilled  water.     Weigh  very 
carefully  the  two  which  have  the  acid  and  then  connect  them 
to  an  aspirator  with  the  water  vessel  between  them.     After 
a  gentle  stream  of  air  has  passed  through  for  a  considerable 
time,  disconnect  and  weigh  the  sulphuric  acid  vessels.     The 
ratio  of  the  gains  in  weight  will  obviously  be  the  relative 
humidity.     Observe  also  the  temperature  of  the  air. 

If  the  actual  aqueous  vapor  pressure  is  determined,  the 
relative  humidity  is  the  ratio  of  this  actual  vapor  pressure 
to  the  saturated  vapor  pressure  corresponding  to  the  actual 
temperature  (Table  LVII) .  The  dew  point  is  the  temperature 
at  which  the  actual  vapor  pressure  is  the  saturated  vapor 
pressure.  If  the  dew  point  is  determined,  the  actual  vapor 
pressure  is  the  saturated  vapor  pressure  corresponding  to  the 
dew  point  (Table  LVII) .  If  the  relative  humidity  is  determined 
the  actual  vapor  pressure  is  the  saturated  vapor  pressure  for 
the  observed  temperature  multiplied  by  the  relative  humidity. 

MEASUREMENT  OF  TEMPERATURE. 
Calibration  of  Mercury  Thermometer. 

45.  Testing    Zero    Point. — A    calorimeter   consisting  of    a 
small  copper  vessel  inside  of  a  larger  is  suitable  for  holding  the 
ice.     Both  vessels  should  be  washed  in  ordinary  tap  water. 


32  INTRODUCTION. 

The  space  between  the  two  vessels  should  be  filled  with  cracked 
ice,  and  the  inner  vessel  filled  with  cracked  ice  and  then 
distilled  water  poured  in  until  the  vessel  is  filled  to  the  brim. 
The  thermometer,  having  been  washed  clean,  is  inserted  in 
the  inner  vessel,  just  enough  of  the  stem  being  exposed  to 
admit  of  the  zero  being  observed.  When  the  reading  has  fallen 
to  i°  the  reading  should  be  observed  every  minute  until  it  is 
stationary  for  five  minutes.  This  stationary  temperature, 
read  to  .1  of  the  smallest  division,  is  the  true  zero  point. 

46.  Testing  Boiling  Point. — The  form  of  boiler  used  for  this 
test  consists  of  a  vessel  for  boiling  water  surmounted  by  a 
tube  up  which  the  steam  passes,  this  tube  being  enclosed  in 
another  down  which  the  steam  passes  to  an  exit  tube  and  a 
pressure  gauge,  Half  fill  the  lower  part  of  the  vessel  with 
water.  Push  the  thermometer  to  be  tested  through  a  cork  in. 
the  top  until  the  boiling  point  is  only  a  degree  or  two  above 
the  cork,  but  take  care  that  the  bulb  of  the  thermometer  does 
not  reach  down  to  the  water.  Apply  heat,  adjusting  it  care- 
fully as  boiling  begins,  so  that  the  pressure  inside,  as  indicated 
by  the  pressure  gauge,  shall  not  materially  exceed  atmospheric 
oressure.  Some  excess  is,  of  course  necessary,  if  there  is  to 
be  a  free  flow  of  steam.  What  excess  is  permissible  may  be 
deduced  from  the  consideration  that  a  rise  of  pressure  of  i  cm. 
(of  mercury  column)  corresponds  to  a  rise  of  boiling  point  of 
.373°.  If  water  is  used  in  the  pressure  gauge,  a  pressure  of 
i  cm.  of  water  column  would  correspond  to  only  .03°  rise  of 
steam  temperature. 

Correction  Table. — Let  /'  Dreading  in  steam  and  let  /  =true 
boiling  temperature  as  given  in  Table  VII  for  the  observed 
barometric  height.  If  t0  is  the  observed  reading  in  ice 
water,  the  true  value  of  each  degree  is 


The  true  temperature  corresponding  to  a  reading  /"  is  therefore 

(t"-Qr 


MEASUREMENT    OF    TEMPERATURE. 


33 


A  table  of  corrections  will  be  found  very  convenient.  For 
every  five  degrees,  for  instance,  the  true  temperature  is  calcu- 
lated and  the  nominal  temperature  is  subtracted.  The 
differences  are  the  corrections  which  must  be  added. 

TABLE  VII. 

Boiling  Temperature  of  Water,  t,  at  Barometer  Pressure,  p  (mm.). 


P- 

t. 

P: 

t. 

P- 

t. 

740 

I  ! 
99.26°               750                       99-630 

760 

100.00° 

41 

.29               51                     .67               61 

.04 

42 

•33                52                      .70               62 

.07 

43 

•37               53 

•74               63 

.1  1 

44 

•4i               54 

.78 

64 

•J5 

45 

•44               55                      -82 

65 

.18 

46 

.48               56                      .85               66 

.22 

47 

•52               57                     -89              67 

.26 

48 

•56               58                     -93 

68 

•29 

49 

•59               59                     -96              69 

•33 

750 

99.63°          760               100.00°          770 

100.36° 

47.  The  Beckmann  Thermometer. — The  Beckmann  ther- 
mometer is  used  for  determining  changes  in  temperature. 
The  bulb  is  large  and  the  stem  is  small  so  that  a  small  change 
of  temperature  is  shown  by  a  large  change  in  reading.  The 
amount  of  mercury  may  be  varied,  and  the  temperature 
corresponding  to  a  particular  reading  will  vary  with  the  amount 
of  mercury  in  the  bulb  and  stem.  There  is  a  reservoir  at  the 
end  of  the  stem  into  which  surplus  mercury  may  be  driven 
by  warming  the  bulb.  A  gentle  jar  will  detach  the  mercury 
in  this  reservoir  when  sufficient  has  been  expelled.  If  one 
desires  to  study  high  temperature  changes,  the  bulb  is  warmed 
until  the  thread  of  mercury  extends  to  the  reservoir,  when 
the  mercury  in  the  reservoir  is  joined  to  it.  The  bulb  is  then 
allowed  to  cool  until  sufficient  mercury  has  been  drawn 
over,  when  the  thread  is  detached  from  the  mercury  in  the 
reservoir  by  a  gentle  jar.  Several  trials  are  often  necessary 
before  the  proper  amount  of  mercury  is  secured. 

In  an  improved  type  of  Beckmann  thermometer,  two 
reservoirs  are  provided,  and  the  first  has  a  scale  which  tells 


34  INTRODUCTION. 

the  amount  of  mercury  required  in  that  reservoir  for  different 
ranges  of  temperature. 

A  thermometer  should  be  gently  tapped  before  a  final  read- 
ing, since  the  mercury  may  adhere  slightly  to  the  glass. 

ELECTRICAL  THERMOMETERS. 

48.  The  most  accurate  thermometer  for  work  below  1000° 
is  the  platinum  resistance  thermometer  which  is  described  under 
Experiment    XXV1.     A   bolometer   is    a    platinum    resistance 
thermometer  with  a  very  thin  blackened   platinum  strip  in 
place  of  the  coil.     It  is  used  for  measuring  the  intensity  of 
the  radiations  which  fall  upon  it. 

49.  The     thermocouple     is     a     simpler     instrument.     For 
temperatures  below  about  300°  a  couple  composed  of  copper 
and  constantin2  wire  is  excellent.     The  two  wires  are  soldered 
together  with  hard  solder,  and  the  junction  is  placed  where 
the  temperature  is  to  be  determined.     The  other  end  of  the 
constantin  wire  is  soldered  to  a  copper  wire  and  this  junction 
is    preferably    kept    in    ice-water.     The    electromotive    force 
developed  is  preferably  measured  by  a  potentiometer  method 
(§73).     It  is  often  simpler  and -more  expeditious  to  connect 
the  thermocouple  directly  to  a  galvanometer   through  a  key 
and  high   resistance    (see  Fig.  57).     The   galvanometer  with 
resistance  is  calibrated  by  applying  a  small   known  electro- 
motive force  (§77),  and  finding  the  value  in  volts  of  one  divi- 
sion on  the  scale.     For  higher  temperatures,  a  couple  composed 
of  platinum,  and  platinum  alloyed  with  10%  of  rhodium  is 
excellent. 3 

The  calibration  curve  of  a  thermocouple — that  is,  a  curve 
giving  the  temperature  corresponding  to  different  electro- 
motive forces — should  be  constructed  by  finding  the  electro- 
motive force  when  one  junction  is  in  ice-water  and  the  other 

1  See  also  Bui.  Bu.  Standards,  1907,  4,  p.  641 ;  1909,  p.  467. 

2  "Constantin"  wire  (60%  Cu.,  40%  Ni)  is  sold  by  the  Driver-Harris  Co.,  of 
Newark,  N.  J.,  under  the  trade  name  of  "Advance"  wire. 

3  Excellent  platinum  thermocouples  are  obtainable  from  Hereaus,  of  Hanau, 
Germany,    or   his   American    agent,    Charles    Engelhardt,    Hudson    Terminal 
Building,  New  York  City. 


ELECTRICAL    THERMOMETERS.  35 

junction  is  in  steam,  and  also  when  the  junction  is  at  some 
other  definitely  known  temperature.  The  boiling  point  of 
sulphur  444.7°  (76  cm.)  is  often  convenient.  Table  XLII 
gives  the  melting  points  of  the  more  common  metals.  The 
electromotive  force  of  a  copper-constantin  thermocouple 
is  about  40  micro-volts  per  degree  difference  in  temperature 
of  the  junctions,  and  that  of  a  platinum,  platinum-rhodium 
thermocouple  is  about  10  micro- volts. 

Radiation  Thermometers. — Optical  pyrometers  are  described 
in  §§212-214,  and  in  Experiment  XXV. 

50.  Beckmann  Apparatus  for  Determining  the  Freezing 
Point  of  a  Solution. — The  apparatus  is  somewhat  delicate  and 
should  be  handled  with  great  care,  particularly  the  Beckmann 
thermometer.  Adjust  the  thermometer  until  it  reads  about 
i°  at  the  melting  point  of  the  pure  solvent.  A  stirrer  and  the 
Beckmann  thermometer  are  in  a  special  test-tube  which  stands 
in  a  larger  test-tube,  which  in  turn  stands  in  a 
cooling  bath  (see  Fig.  7).  The  solution  should, 
if  possible,  be  undercooled  by  placing  the  test- 
tube  with  the  solution  in  the  outer  bath,  which 
should  be  several  degrees  below  the  expected 
freezing  point.  The  test-tube — the  outside  hav- 
ing been  wiped  dry — is  quickly  transferred  to 
the  larger  test-tube,  and  both  solution  and  outer 
liquid  are  stirred.  If  the  solution  has  been 
undercooled,  the  temperature  will  rise,  when 
freezing  begins,  and  the  highest  steady  temperature  attained 
should  be  recorded. 


Measurement  of  Boiling  Point  of  a  Solution. 

51.  Beckmann  Apparatus. — There  are  two  common  forms 
of  boiling  point  apparatus.  In  both  the  solution  (about  20 
c.c.)  is  contained  in  a  special  boiling  tube,  with  glass  beads, 
garnets  or  platinum  tetrahedra  at  the  bottom.  The  solution 
entirely  covers  the  bulb  of  a  Beckmann  thermometer.  The 
latter  should  not  touch  the  glass  beads,  etc.  In  one  form  of 


INTRODUCTION. 


apparatus,  this  boiling  tube  is  heated  from  beneath  by  a 
special  small  flame  and  is  well  protected  by  gauze  beneath, 
and  around  the  sides  by  two  glass  cylinders  covered  by  a  mica 
cap. 

In  another  form  the  boiling  tube  is  surrounded  by  a  vapor 
jacket.  With  the  latter,  both  boiling  tube  and  jacket  are 
provided  with  spiral  air  condensers  for  condensing  the  vapor. 
The  first  type  (Fig.  8)  uses  a  small 
internal  water  condenser,  to  which 
water  is  supplied  from  a  reservoir  at  a 
small  elevation. 

The  Beckmann  thermometer  should 
be  so  adjusted  that  it  reads  about  i° 
in  the  vapor  of  the  boiling  solvent. 
For  directions  for  adjustment  see  §47. 

If  a  vapor  jacket  is  used,  it  should 
contain  about  40  c.c.  of  solvent,  which 
will  probably  have  to  be  replenished 
from  time  to  time.  Use  a  small  flame 
or  flames,  and  even  when  the  inner 
liquid  boils,  the  boiling  should  be  so 
gentle  that  only  one  drop  in  about  10  sec.  falls  from  the  con- 
denser. Gently  tap  the  thermometer  before  reading. 

52.  Landesberger- Walker  Apparatus. — In  this  method  for  de- 
termining the  boiling  point,  the  pure  solvent  is  boiled  rather 
than  the  solution.  The  vapor  from  the  pure  solvent  in  the 
boiling  vessel  (Fig.  8a)  is  largely  condensed  in  the  solution 
contained  in  B.  What  is  not  condensed  escapes  through  a 
small  hole,  (a),  and  condenses  and  collects  in  the  outer  vessel, 
C,  or  is  condensed  in  the  attached  condenser.  The  latent 
heat  of  the  vapor  raises  the  temperature  of  the  solution  in  B 
to  its  boiling  point.  When  this  state  is  attained  the  tempera- 
ture as  read  on  the  tenth  degree  thermometer  T  becomes 
approximately  constant.  B  is  first  filled  with  enough  of  the 
pure  solvent  to  cover  the  bulb  of  the  thermometer,  and  its 
boiling  point  is  carefully  determined  to  hundredths  of  a  degree 
by  estimating  tenths  of  the  thermometer  divisions.  Repeat 


FIG. 


BOILING    POINT   APPARATUS. 


37 


twice.  The  excess  of  the  solvent  which  has  condensed  in  B 
and  also  what  is  in  C  is  then  poured  into  the  boiling  vessel,  and 
a  very  carefully  weighed  amount  of  the  solute  is  placed  in  B. 
The  boiling  point  is  redetermined  by  passing  vapor  into  the 
solution  in  B  until  the  temperature  becomes  constant. 


FIG.  8a. 


Immediately  after  constancy  is  obtained,  the  delivery  tube 
and  thermometer  are  removed  from  the  solution  and  its 
volume  is  carefully  read  from  the  graduations  on  the  outside 
of  the  vessel.  Reassemble  the  apparatus,  start  again  the  flow  of 
vapor,  and  redetermine  the  boiling  point  and  the  volume.  If 
possible,  repeat  a  second  time.  If  the  solvent  is  inflammable, 
the  flame  must  be  removed  or  put  out  before  the  vessel  B  is 
opened. 

If  garnets,  glass  beads,  or  fresh  tile  are  placed  in  the  boiling 
vessel,  the  boiling  will  be  more  regular. 


38  INTRODUCTION. 

CALORIMETERS. 

53.  Fig.  9  represents  Berthelot's  calorimeter  which,  with 
modifications  for  special  purposes,  is  suitable  for  most  heat 
experiments.  The  inner  calorimeter  consists  of  a  highly 
polished  vessel  of  thin  metal  (or  glass,  if  the  liquid  attacks 
metals)  with  a  cardboard  cover,  through  which  pass  the 


FIG.  9. 

thermometer  and  the  stirrer,  and  which  supports  any  other 
necessary  bodies.  It  rests  on  a  cardboard  cross  formed  by 
cutting  half  through  two  pieces  of  cardboard  and  fitting  them 
together  at  the  cuts.  This  cross  sets  in  an  outer,  highly 
polished  vessel  which,  in  turn,  is  surrounded  by  a  large  water 
jacket.  The  intermediate  vessel  is  closed  by  a  second  card- 
board cover.  If  the  temperature  changes  are  small,  an  outer 


CALORIMETERS.  39 

wooden  casing  serves  almost  as  satisfactorily  as  the  water 
jacket.  It  is  very  essential  that  the  radiation  correction 
(§59)  be  determined  with  great  care. 

54.  The  water  equivalent  of  a  simple  calorimeter  is  equal  to 
the  sum  of  the  water  equivalents  (mass  X  specific  heat)  of 
the  inner  calorimeter  vessel,  stirrer,  thermometer  bulb,  etc. 
The  specific  heats  of  the  more  common  substances  are  given  in 
Tables  LI  1 1— LV.     The   specific   heat   of  glass   and   mercury 
may  be  taken  as  .47  per  cubic  centimeter. 

Combustion  Calorimeters. 

55.  Constant  Volume  Calorimeter    (Hempel    Bomb  Calor- 
imeter).— A  pellet  of  the  substance  to  be  consumed  is  formed  in 
a  press,  a  cotton  cord  being  imbedded  with  a  loose  end.     After 
being  pared  down  to  about  i  gr.  and  brushed,  it  is  carefully 
weighed.  .  It  is  then  suspended  in  a  Hempel  combustion  bomb, 
and  the  thread  is  wrapped  around  a  platinum  wire  connecting 
the  platinum  supports  of  the  basket.     The  terminals  attached 
to  these  supports  are  later  connected  (through  a  key)  with 
sufficient   Edison   or  storage  cells  to  just  bring  the  wire  to  a 
brilliant  incandescence  (as  ascertained  by  a  preliminary  trial) . 

The  bomb  is  charged  with  oxygen  under  at  least  fifteen 
atmospheres  pressure  either  from  a  charged  cylinder  or  pro- 
duced by  a  retort.  Bomb  and  pressure  gauge  should  be 
immersed  in  water  while  the  oxygen  is  being  supplied.  Ascer- 
tain that  the  bomb  valve  is  open  and  that  all  connections  are 
screwed  tight.  Open  the  cylinder  valve  (if  a  cylinder  is  used) 
until  the  pressure  becomes  high,  and  then  close.  Lift  the 
bomb-out  of  the  water,  loosen  one  of  the  connections,  and  allow 
the  mixture  of  air  and  oxygen  to  escape;  then  tighten,  replace 
in  water,  open  the  cylinder  valve  again  until  the  pressure  be- 
comes high ;  close  both  the  cylinder  valve  and  that  of  the  bomb, 
and,  finally,  disconnect  and  dry  the  bomb.  (If  the  oxygen  is 
produced  in  a  retort,  partly  fill  the  latter  with  a  five-to-one 
mixture  of  potassium  chlorate  and  manganese  dioxide,  connect 
to  the  bomb  and  pressure  gauge,  and  heat  the  upper  par 
of  the  retort  slowly  with  a  Bunsen  burner.) 


4O  INTRODUCTION. 

Attach  the  electric  terminals,  place  the  bomb  in  the  special 
vessel  containing  about  a  litre  of  water,  adjust  the  Beck- 
mann  thermometer  to  read  about  i°  (see  §47),  stir  the  water 
continually,  and  read  its  temperature  every  half-minute 
for  five  minutes,  estimating  to  tenths  of  the  smallest  division. 
Close  the  electric  switch;  after  a  few  seconds  open  it  and  read 
the  temperature  of  the  continually  stirred  water  for  ten 
minutes. 

Let  q  be  the  heat  value  of  the  substance  and  m  the  mass  of 
the  specimen,  mw  the  mass  of  the  water,  e  the  water  equiva- 
lent of  the  bomb,  ^  the  initial  temperature,  and  t2  the  final 
temperature  (corrected  for  radiation,  see  §59).  Then 


(cal.pergr.) 


The  best  method  in  practice  to  determine  e  is  to  repeat  the 
determination,  using  salicylic  acid  as  fuel,  and  assuming  its 
heat  value  to  be  5300  calories  per  gram.1 

56.  Constant  Pressure  Calorimeter  (Rosenhain's  Calor- 
imeter2) .  —  Instead  of  burning  the  substance  in  a  fixed  volume 
of  highly  compressed  oxygen,  the  oxygen  is  supplied  continu- 
ously at  only  slightly  above  atmospheric  pressure. 

The  substance  is  pulverized,  and  a  sample  is  compressed, 
in  a  special  screw  press,  into  a  pellet  weighing  about  one 
gram.  This  is  placed  on  a  porcelain  dish  which  rests  on  the 
bottom  of  the  inside  chamber.  The  ignition  wire  should  be 
about  3  cm.  of  No.  30  platinum  wire  and  the  external  terminals 
should  be  connected  to  storage  battery  terminals  through  a 
key  and  a  resistance  such  that  the  wire  will  glow  brightly. 
A  tank  is  charged  with  oxygen  from  a  cylinder  or  generated 
from  "oxone"  and  water.  The  action  of  the  different  valves 
having  been  studied,  the  apparatus  should  be  assembled, 
the  upper  side  valve  (Fig.  10)  being  closed  and  the  ball 

1  Jaeger  and  Steinwehr  have  shown  that  the  most  accurate  method  of  deter- 
mining e  is  to  use  a  heating  coil  on  the  outside  of  the  calorimeter.     Ann.  der 
Phys.  1906,  21,  p.  23. 

2  This  excellent  calorimeter  which  is  very  satisfactory  for  students'  laboratory 
use,  is  made  by  the  Cambridge  Scientific  Instrument  Co.,  Cambridge,  England. 


CALORIMETERS. 


LJ 


valve,  lowered.  Connect  with  the  oxygen  supply  through  a 
wash-bottle,  turn  on  a  very  gentle  stream  of  oxygen,  and 
pour  into  the  outer  vessel  1500  c.c.  of  water  at  the  room 
temperature.  The  water  should  just  cover  the  combustion 
chamber. 

If  a  Beckmann  thermometer  (see  §47)  is  used,  adjust 
to  read  between  o°  and  i°  in  this  water.  The  bulb  should  be 
supported  on  a  level  with  the  center  of  the 
combustion  chamber.  Read  the  tempera- 
ture every  half-minute  for  five  minutes, 
then  increase  the  oxygen  current,  and  care- 
fully read  the  temperature  and  the  time, 
close  the  key  and  ignite  the  pellet  with  the 
hot  platinum  wire,  and  immediately  remove 
the  wire.  During  such  operations  it  is  best 
to  hold  the  inner  vessel  steady  by  grasping 
the  oxygen  inlet  tube.  Keep  the  water 
pressure  in  the  gasometer  constant  and  as 
combustion  proceeds,  increase  the  flow  of 
oxygen.  Read  the  thermometer  every  FIG.  10. 

minute. 

When  combustion  has  ceased,  move  the  hot  wire  about  to 
ignite  any  unconsumed  particles.  Keep  the  wire  hot  as  short 
a  time  as  possible  and  remove  it  immediately  from  any  com- 
bustion, otherwise  it  is  liable  to  be  melted. 

Finally,  turn  off  the  oxygen  supply,  open  the  upper  valve, 
and  raise  the  ball  valve,  allowing  the  water  to  fill  the  inner 
chamber.  Then  force  out  the  water  by  closing  both  valves 
and  turning  on  the  oxygen.  Record  the  highest  tempera- 
ture and  the  time,  and  the  temperature  every  half  minute  for 
five  minutes.  For  the  radiation  correction,  see  §59  and  for 
the  formulas,  see  the  preceding  section. 

To  determine  e,  assemble  the  apparatus,  including  the 
Beckmann  thermometer,  and  pour  in  1000  c.c.  of  water. 
Determine  very  carefully  the  temperature  with  a  0.1°  ther- 
mometer and  then  add  500  c.c.  of  water  at  about  50°,  the 
temperature  of  which  has  also  been  very  carefully  determined. 


INTRODUCTION. 


Determine  also  very  carefully  the  final  steady  temperature. 
From  these  data  calculate  e. 

Add  to  anthracite  coal  one-third  its  weight  of  cane  sugar. 
(Heat  of  combustion  of  sugar  =3900  calories  per  gram.) 

57.  Heat  Value  of  a  Gas  (Junker's  Calorimeter). — A 
measured  volume  (v  liters)  of  gas  under  an  observed  pressure, 
p,  is  burned  in  the  calorimeter,  and  the  rise  of  temperature, 
from  tj°  to  ta°  of  a  mass  of  m  gr.  of  water,  is  determined. 
The  flow  of  water  and  gas  is  so  regulated  that  the  burned 

gas  leaves  the  calorimeter,  at  ap- 
proximately the  temperature  of  the 
entering  gas,  and  there  should  be 
a  difference  of  at  least  6°  in  the 
temperature  of  the  in-  and  out- 
flowing water.  Also,  the  flow  of 
water  must  be  sufficient  to  furnish 
a  constant  small  overflow  at  the 
supply  reservoir  (Fig.  n).  The 
burner  should  be  lighted  outside  of 
the  calorimeter. 

When  the  temperatures  indi- 
cated on  the  various  thermome- 
ters have  become  constant,  note 
the  gasometer  reading,  and  imme- 
diately collect  in  graduates  the 

heated  overflowing  water,  and  also  the  water  condensed  by 
the  combustion  of  the  gas.  Let  the  mass  of  the  latter  be  m' 
gr.,  and  its  temperature  t'°.  Note  the  temperatures  of  the 
inflowing  and  outflowing  water  every  15  sec.  until  two  or 
three  liters  have  passed  through.  Then  immediately  note  the 
gasometer  reading,  and  remove  the  graduates. 

Assuming  that  condensation  of  the  gas  occurs  at  100°, 
the  heat  liberated  is  m'  (537 +100  — £')•  If  q  represents  the 
heat  value  of  the  gas  in  gram-calories  per  liter,  and  v  the 
volume  reduced  to  o°,  760  mm., 

mfe-f,)— m'  (637-0 
v 


FIG.  ii. 


RADIATION    CORRECTION. 


43 


58.  Heat  Value  of  a  Liquid. — To  determine  the  heat  value  of 
a  liquid  fuel,  the  burner  (§57)  is  replaced  by  a  suitable  lamp 
which  is  attached  to  one  arm  of  a  balance.  The  rate  at  which 
the  liquid  is  consumed  is  determined  from  the  weights  in  the 
pan  on  the  other  side  at  different  times.  It  is  best  to  make 
the  weight  in  the  pan  slightly  deficient  and  note  the  exact 
time  when  the  balance  pointer  passes  zero  as  the  liquid  is 
consumed.  Practically  complete  combustion  is  obtained 
with  a  " Primus"  burner,  supplied  from  a  reservoir  where  the 
liquid  is  under  considerable  pressure.  With  very  volatile 
liquids,  the  opening  of  the  burner  must  be  large  and  the  pre- 
heating tubes  must  be  in  the  cooler  part  of  the  flame. 


TO  Tt  T2  T3 

FIG.'  12. 

RADIATION  CORRECTION. 

59.  Fig.  12  represents,  on  an  exaggerated  scale,  typical 
observations  during  a  calorimeter  experiment.  The  calor- 
imeter and  contents  are  cooled  initially  below  the  temperature 


44  INTRODUCTION. 

of  the  room.  The  temperature  is  read  at  regular  intervals 
before  the  heating  commences,  during  heating,  and  during  the 
subsequent  cooling. 

The  initial  rate  of  rise  of  temperature 


r,-r« 


is  determined.     At  time  7\  heating  begins  and  continues  till  time  T2. 
The  final  temperature  gradient 


T3-T 

is  then  determined  by  regular  observations  of  the  temperature  after 
the  maximum  is  reached.  During  the  time  T—T^  the  calorimeter 
was  gaining  heat  from  the  room  as  well  as  from  the  source  in  the 
calorimeter.  By  Newton's  law  of  cooling,  the  rate  of  change  of 
temperature  is  proportional  to  the  difference  of  temperature.  If  t' 
is  the  average  difference  of  temperature  between  the  calorimeter  and 
the  room  during  this  time,  the  average  rate  of  warming  from  the 
room  is 


and  the  resulting  gain  of  temperature  is 


If  the  curve   B  C  is  approximately  a  straight  line,  this  is  equal  to 


Similarly,  the  loss  of  temperature  during  the  time  T2—  T  is 

t" 
t2—t 

where  t"  is  the  mean  excess  above  the  room  temperature.      Therefore, 
the  corrected  rise  of  temperature  is 


t'  and  t"  can  easily  be  estimated  from  such  a  plot  of  the  temperature 
readings.  If  B  C  D  is  approximately  a  straight  line,  then  the  above 
expression  reduces  to 


RADIATION    CORRECTION.  45 

If  we  agree  to  call  a  rise  of  temperature  positive,  this  expression 
becomes 


(23) 


Or,  the  corrected  rise  of  temperature  is  the  apparent  rise  in 
temperature ;  less  the  average  rate  of  rise  of  temperature  due  to 
radiation,  multiplied  by  the  time  of  heating.  An  analogous 
expression  will  give  the  correction  if  there  is  an  absorption 
rather  than  an  emission  of  heat  in  the  calorimeter.  These 
approximate  equations  are  particularly  useful  when  the 
temperature  of  the  room  is  indefinite. 

Measurements  in  Light. 

60.  Monochromatic  Light. — The  simplest  and  most  useful 
monochromatic  light  is  the  sodium  flame.  Sodium  may  be 
introduced  into  a  bunsen  flame  by  surrounding  the  tube  of 
the  burner  with  a  tightly  fitting  cylinder  of  asbestos  which 
has  been  saturated  with  a  strong  solution  of  common  salt  and 
formed  into  cylindrical  shape  by  wrapping  around  the  burner 
while  still  damp.  As  the  top  of  the  cylinder  is  exhausted, 
it  is  torn  off  and  the  rest  of  the  tube  is  pushed  up  into  the 
lower  part  of  the  flame.  A  piece  of  hard  glass  tubing  held 
in  the  flame  will  also  give  a  good  sodium  light. 

Elements  giving  red,  green,  blue  and  violet  light  will  be 
found  in  Table  LXII.  Salts  of  these  elements  (e.g.,  KNO3, 
SrCl2,  CaCl2,  LiCl)  may  be  introduced  into  the  outer  edge  of 
a  bunsen  flame,  either  in  a  thin  platinum  spoon,  on  copper 
gauze,  or  by  a  piece  of  wood  charcoal  which  has  absorbed  a 
solution.  If  a  very  intense  light  is  not  required,  a  vacuum 
tube  is  a  very  satisfactory  source  (§215  and  Table  LXII  I). 
Intense  light  of  one  general  color  may  be  obtained  by  filtering 
sun  light  or  the  light  from  an  arc  light  through  colored  glass 
or  gelatine.  The  solutions  given  in  Table  VIII  give  much 
purer  mono-chromatic  light. 


INTRODUCTION. 


TABLE  VIII. 
Light  Filters  (Landolf). 


Color 

Thickness 
of  Layer 
(mm.) 

Aqueous 
Solution  of 

Grams  per 

100  C.C. 

Average 
Wave  Length 
(Angstrom  Units) 

Red 

20 
20 

Crystal  violet  560 
Potassium  chromate 

.005 

10. 

6560 

Green 

20 

20 

Copper  chloride 
Potassium  chromate 

60. 

10. 

533° 

Blue 

20 

20 

Crystal  violet 
Copper  sulphate 

I5:°°s 

The  Spectrometer. 

61.  Spectrometer  Adjustment. — A  spectrometer  consists  of 
a  framework  supporting  a  telescope  and  a  collimator,  which  are 
movable  about  a  vertical  axis,  and  a  platform  movable  about 
the  same  axis.  The  collimator  is  a  tube  containing  an  adjusta- 
ble slit  at  one  end  and  a  lens  at  the  other  end.  The  purpose  of 
the  collimator  is  to  render  light  coming  from  the  slit  parallel. 
Hence,  the  slit  of  the  collimator  should  be  in  the  principal 
focus  of  the  lens. 

The  telescope  is  for  the  purpose  of  viewing  the  light  that 
comes  from  the  collimator,  either  directly  or  after  the  light 
has  been  refracted,  reflected,  or  diffracted.  Hence,  since  the 
light  that  comes  from  the  collimator  is  supposed  to  be  parallel- 
that  is,  as  if  it  comes  from  a  very  distant  source — it  follows  that 
if  the  telescope  is  to  receive  the  light  and  form  a  distinct 
image  of  the  slit,  the  telescope  must  be  focused  as  for  a  very 
distant  object  (theoretically  an  infinitely  distant  one).  The 
first  adjustment  is  to  focus  the  telescope.  First  focus  the 
eye-piece  of  the  telescope  on  the  cross  hairs  and  then  focus 
the  whole  telescope  on  the  most  distant  object  visible.  The 
telescope  will  now  be  in  focus  for  parallel  rays.  Turn  the 
telescope  to  view  the  image  of  the  slit  formed  by  the  collimator 
and  adjust  the  slit  until  its  image  is  seen  most  distinctly. 

TMann,  "Manual  of  Advanced  Optics,"  p.  185. 


SPECTROMETER.  47 

62.  Measurements  of  the  Angle  of  a  Prism. — Fix  both  tele- 
scope and  collimator  and,  rotating  the  platform  upon  which 
the  prism  is  mounted,  find  the  positions  of  the  prism  and 
platform  in  which  •  the  light  is  reflected  from  the  two  faces 
inclosing  the  angle.     The  supplement  of  the  angle  between 
these  two  positions  is  the  angle  of  the  prism. 

63.  Minimum     Deviation. — The     position     of     minimum 
deviation  (Fig.  13)  is  such  that  the  image  of  the  slit  seen  in  the 
telescope  moves  in  the   same   direction    (that  of  increasing 
deviation)  no  matter  which 

way  the  platform  carrying 
the  prism  is  turned.  There 
are,  of  course,  two  positions 
in  which  the  deviation  can 
be  obtained,  one  with  the 
refracting  edge  pointing  to 

the  right  of  the  observer,  and  the  other  with  it  pointing 
to  the  left.  The  deviation  in  each  case  is  the  angle  between 
the  corresponding  position  of  the  telescope  and  its  position 
when  looking  directly  into  the  collimator,  the  prism  being 
removed.  But  it  is  not  necessary  to  remove  the  prism,  for 
it  is  easily  seen  that  the  minimum  deviation  must  also  be 
equal  to  half  of  the  angle  between  the  two  positions  of  the 
telescope  when  observing  the  minimum  deviation.  From  A, 
the  angle  of  the  prism  and  D,  the  angle  of  the  minimum 
deviation,  the  index  of  refraction  may  be  calculated  by  the 
formula 


.     A+D 

sin 


sn  - 
2 


(24) 


Descriptions  of  the  two  principal  types  of  refractometer  are 
given  in  §§220,  221.  The  most  common  types  of  polarimeter 
are  described  in  §§230-233. 


48  INTRODUCTION. 

Wave  Length  of  Light  by  Diffraction  Grating. 

64.  A  diffraction  grating  consists  of  a  great  many  lines  ruled 
parallel  and  equidistant  on  a  plane  (or  concave)  surface.  If 
the  surface  be  that  of  glass,  the  grating  is  a  transmission  grat- 
ing; if  of  metal,  a  reflection  grating.  If  a 
transmission  grating  be  placed  perpendicular 
to  homogeneous  parallel  light  from  a  colli- 
mator  (see  §61)  and  with  the  lines  parallel 
to  the  slit,  a  series  of  spectra  will  be  formed 
on  either  side  of  the  beam  of  light  which  is 
transmitted  without  deviation  (Fig.  13 a). 
If  n  =  the  number  or  order  of  a  particular 
spectrum  counting  from  the  center,  <j>  =  the 
deviation  or  angle  that  the  rays  forming  the 
spectrum  make  with  the  original  direction  of 
the  light,  a  =  the  grating  space  or  average  distance  between  the 
centers  of  adjacent  lines,  and  X  =the  wave-length  of  the  light 

X  =  -a  sin  d  (25) 

The  slit  of  the  collimator  is  made  accurately  vertical.  The 
grating  is  mounted  on  the  platform  of  a  spectrometer  so  that 
the  lines  are  vertical  and  so  that  the  plane  of  the  lines  includes 
the  axis  of  rotation  of  the  instrument  and  is  perpendicular  to 
the  axis  of  the  collimator.  The  lines  are  parallel  to  the  slit  when 
the  spectrum  of  some  homogenous  light  (e.g.,  from  a  sodium 
flame)  is  as  distinct  as  possible.  When  the  plane  of  the  grating 
is  perpendicular  to  the  incident  light,  the  deviation  (on  opposite 
sides)  of  the  two  spectra  of  the  same  order  should  be  equal. 
This  adjustment  is  also  secured  when  that  part  of  the  beam 
which  is  reflected  back  to  the  collimator  appears  co-axial  with 
its  object-glass. 

For  a  final  measurement  of  the  deviation  of  any.  portion  of  a 
spectrum  the  mean  of  at  least  three  observations  on  each 
side  should  be  taken.  0  is  half  the  mean  angle  between  the 
positions  on  the  two  sides.  Analogous  directions  apply  to  a 
reflection  grating. 


ELECTRICAL    MEASUREMENTS.  49 

If  the  grating  space  be  not  too  small  it  may  be  obtained  by 
measurements  with  a  micrometer  microscope .  (§27.)  Secure 
the  best  possible  illumination  of  the  lines.  Set  the  cross 
hair  of  the  microscope  on  a  line  and  read  the  position  of  the 
divided  head  (circular  scale) .  Watching  the  lines  through  the 
microscope,  turn  the  screw,  always  in  the  same  direction, 
until,  for  example,  the  tenth  line  is  under  the  cross  hair, 
and  read  the  circular  scale.  Then  turn  the  screw  until 
the  tenth  line  from  this  is  under  the  cross  hair,  read  the  scale, 
and  so  on.  Take  ten  such  groups  in  different  parts  of  the 
grating.  From  the  mean  find  the  average  grating  space. 

When  the  grating  space  is  very  small,  the  wave-length  of 
some  well-known  spectrum  (e.g.,  sodium)  is  assumed  (Table 
LXIII)  and  the  grating  space  is  derived  by  reversing  the  process 
of  finding  the  wave-length. 

Electrical  Measurements. 

65.  Resistance. — The  resistance  of  a  conductor  is  determined 
by  comparison  with  another  conductor  whose  resistance  is 
known.  Standard  resistances  can  be  bought  of  several  makers 
with  a  guaranteed  accuracy  of  .02%,  as  certified  by  one  of  the 
National  Laboratories. 

A  resistance-box  consists  of  a  number  of  known  resistance 
coils  joined  so  that  each  one  bridges  the  gap  between  two 
of  a  series  of  brass  blocks,  placed  in  line  on  the  cover  of  the 
box  in  which  the  coils  are  suspended.  For  each  gap  a  plug  or 
connector  is  also  provided,  and  when  the  plug  is  inserted  into 
the  gap  the  resistance  at  the  gap  is  "cut  out"  or  practically 
reduced  to  zero.  The  coils  are  wound  so  as  to  be  free  from 
self-induction. 

Before  beginning  work,  it  is  advisable  to  clean  the  plugs 
with  fine  emery-cloth  so  that  they  may  make  good  contacts, 
and  thereafter  care  should  be  taken  not  to  soil  them  with  the 
fingers.  If  any  of  the  plugs  are  in  loosely,  there  will  be  ap- 
preciable resistance  at  the  contact.  Hence,  every  plug  should 
be  screwed  in  firmly,  but  not  violently.  When  any  one  plug 
has  been  withdrawn,  the  others  should  be  tested  before 


50  INTRODUCTION. 

proceeding,  for  the  removal  of  one  may  loosen  the  contact  of 
the  others.  This  precaution  is  especially  important  in  making 
a  final  determination. 

66.  Wheatstone's  Bridge. — The  unknown  resistance  is 
usually  compared  with  a  known  resistance  by  means  of  the 
arrangement  of  conductors  known  as  Wheatstone's  bridge,  the 
elementary  principles,  construction,  and  manipulation  of 
which  is  undoubtedly  familiar  to  the  student. 


I, 

-    h 

B 

12 

FIG    14, 

The  simplest  type  of  bridge  is  the  slide  wire  bridge,  illustrated 
in  Fig.  14.  The  known  resistance  R  is  adjusted  until  the  slide, 
B,  is  as  near  as  possible  to  the  center  of  the  uniform,  high 
resistance  wire  A  C,  when  the  galvanometer  deflection  is 
a  minimum.  If  /x  and  12  are  the  lengths  of  the  two  portions 
of  the  wire,  when  the  deflection  of  the  galvanometer  is  a 
minimum, 

X=lfR  (26) 

The  length  of  the  slide  wire  can  be  greatly  increased  by 
winding  it  in  a  groove  on  a  drum.  The  contact  revolves  around 
the  drum  and  is  advanced  by  a  screw  whose  pitch  is  equal  to 
the  distance  between  the  turns  of  the  wire.  The  length  of  the 
common  slide  wire  is  100  cm.  .'.  /2  =  ioo— /x  Table  IX 
gives  the  value  of 


100— 


for  various  values  of  Zx. 


WHEATSTONE'S  BRIDGE. 


TABLE  IX. 
Meter  Slide  Wire  Bridge  (Obach). 

[Value  of  -     l—r  for  various  values  of  /x.l 

100  — /r 


/I 

o.o 

O.I 

O.  2 

°-3 

0.4 

o-5 

0.6 

0.7 

0.8 

0.9 

30 

0.4286 

4306 

4327 

4347 

4368 

4389 

4409 

443° 

445  1 

4472 

31 

4493 

45J4 

4535 

4556 

4577 

4599 

4620 

4641 

4663 

4684 

32 

4706 

4728 

4749 

4771 

4793 

4815 

4837 

4859 

4881 

4903 

33 

4925 

4948 

497° 

4993 

5°!5 

5038 

5060 

5083 

5106 

5*29 

34 

5J52 

5175 

5198 

5221 

5244 

5267 

5291 

53*4 

5337 

5361 

35 

5385 

5408 

5432 

5456 

548o 

55°4 

5528 

5552 

5576 

5601 

36 

5625 

5650 

5674 

5699 

5723 

5748 

5773 

5798 

5823 

5848 

37 

5873 

5898 

5924 

5949 

5974 

6000 

6026 

6051 

6077 

6103 

38 

6129 

6i55 

6181 

6208 

6234 

6260 

6287 

63  !3 

6340 

6367 

39 

6393 

6420 

6447 

6475 

6502 

6529 

6556 

6584 

6611 

6639 

40 

0.6667 

6695 

6722 

6750 

6779 

6807 

6835 

6863 

6892 

6921 

4i 

6949 

6978 

7007 

7036 

7065 

7094 

7123 

7153 

7182 

7212 

42 

7241 

7271 

730i 

733i 

736i 

7391 

7422 

7452 

7483 

7513 

43 

7544 

7575 

7606 

7637 

7668 

7699 

7731 

7762 

7794 

7825 

44 

7857 

7889 

7921 

7953 

7986 

8018 

8051 

8083 

8116 

8149 

45 

8182 

8215 

8248 

8282 

8315 

8349 

8382 

8416 

8450 

8484 

46 

8519 

8553 

8587 

8622 

8657 

8692 

8727 

8762 

8797 

8832 

47 

8868 

8904 

8939 

8975 

9011 

9048 

9084 

9121 

9*57 

9194 

48 

9231 

9268 

9305 

9342 

9380 

9418 

9455 

9493 

953  1 

957° 

49 

9608 

9646 

9685 

9724 

9763 

9802 

9841 

9881" 

9920 

9960 

50 

I.OOO 

.004 

.008 

.012 

1.016 

.020 

.024 

1.028 

I-°33 

i  037 

5i 

1.041 

•045 

.049 

•053 

1.058 

.062 

.066 

1.070 

I-°7S 

1.079 

52 

1.083 

.088 

.092 

.096 

I.IOI 

.105 

.no 

.114 

1.119 

1.123 

53 

1.128 

.132 

•*37 

.141 

1.146 

!5i 

•155 

.160 

1.165 

1.169 

54 

1.174 

.179 

•183 

.188 

1-193 

.198 

.203 

.208 

1.212 

1.217 

55 

1.222 

.227 

.232 

•237 

1.242 

.247 

252 

•257 

IV262 

1.268 

56 

!-273 

-278 

.283 

.288 

1.294 

•  299 

304 

•3°9 

I-3I5 

1.320 

57 

1.326 

•331 

•336 

•342 

J-347 

•353 

•358 

•364 

J-37° 

J-375 

58 

1.381 

•387 

•392 

.398 

1.404 

.410 

•415 

.421 

1.427 

1-433 

59 

r-439 

•445 

•451 

•457 

1.463 

.469 

•475 

.481 

1.488 

1.494 

60 

1.500 

.506 

•5r3 

•5i9 

!-525 

•532 

•538 

•545 

i-55i 

I-558 

61 

1.564 

•571 

•577 

•584 

I-59I 

•597 

.604 

.611 

1.618 

1.625 

62 

1.632 

•639 

.646 

-653 

i.  660 

.667 

.674 

.681 

1.688 

1.695 

63 

*-1Q3 

.710 

.717 

•725 

!-732 

.740 

•747 

•755 

1.762 

1.770 

64 

1.778 

.786 

•793 

.801 

1.809 

.817 

.825 

•833 

1.841 

1.849 

65 

1-857 

.865 

•874 

.882 

1.890 

.899 

.907 

i-9i5 

1.924 

1-933 

66 

1.941 

•95° 

•959 

.967 

1.976 

•985 

•994 

2.003 

2.012 

2.021 

67 

2  030 

2.040 

2.049 

2.058 

2.067 

2.077 

2.086 

2.096 

2.IO6 

2.II5 

68 

2.125 

2-135 

2-145 

2-155 

2.165 

2-175 

2.18.5 

2-195  ! 

2.2O5 

2.215 

69 

2.226 

2.236 

2.247 

2.257 

2.268 

2.279 

2.289 

2.300 

2.3II 

2.322 

5.2 


INTRODUCTION. 


67.  The  Box  Type  of  Bridge  or  Post  Office  Bridge  will  usually 
be  found  more  accurate  and  convenient.  The  two  lengths  of 
resistance  wire  /x  and  12  are  replaced  by  groups  of  resistances 
(Pand  Q,  Fig.  15)  each  containing  a  i-ohm,  a  zo-ohm,  a  zoo-ohm 
and  a  icoo-ohm  coil.  One  coil  is  used  in  each  group  or  "  arm  " 
and  the  choice  is  such  that  the  known  resistance,  R,  is  as  great 
as  possible.  The  galvanometer  must  be  so  sensitive  that, 


FIG.  15. 


whatever  P  and  Q,  a  change  of  one  ohm  in  the  known  resistance, 
R,  is  detected  by  the  galvanometer.  Since  the  known 
resistance  usually  has  no  coil  smaller  than  one  ohm,  the  error 
from  uncertainty  in  smaller  units  is  less,  the  greater  the  known 
resistance.  If,  for  example,  the  sum  of  all  the  coils  in  R  is 
11,000  and  X  is  less  than  100,  P  had  better  be  1000  and  Q  10. 
If,  however,  X  is  between  1 00,000  and  i  ,000,000,  P  should  be  i  o 
and  Q  1000.  When  the  approximate  value  of  X  has  been 
determined,  suitable  values  are  given  P  and  Q,  and  R  is 
adjusted  until  a  change  of  one  ohm  changes  the  direction  of 


WHEATS-TONE'S  BRIDGE. 


53 


the  deflection  of  the  galvanometer.  P  and  Q  are  called  the 
ratio  coils. 

The  coils  of  a  box  bridge  may  have  appreciable  capacity, 
and  with  alternating  currents  this  reduces  but  does  not 
usually  annul  its  advantages  over  the  slide  wire  bridge. 

68.  A  modification  known  as  the  Callendar  and  Griffith's 
bridge  is  very  satisfactory  for  such  resistances  as  platinum 


FIG.  i 6. 

thermometers  (§48,  Exp.  XXV).  It  is  schematically  illus- 
trated in  Fig.  1 6.  The  two  ratio  arms  P  and  Q  are  single 
equal  coils.  The  known  arm  A  D  and  the  unknown  arm  E  C 
are  joined  by  a  slide  wire  D  E.  The  galvanometer  is  connected 
to  an  exactly  similar  parallel  wire  and  the  two  are  joined  by  a 
sliding  cross-piece,  y,  of  the  same  material  as  the  wires.  R  is 
adjusted  until  a  position  of  y  is  found  for  which  the  galvan- 
ometer shows  no  deflection.  Since  P  =Q,  the  resistance  R  plus 
that  of  D  y  is  then  equal  to  the  resistance  X  plus  that  of  E  y,  or 
X  =  R  +  2  F  y  (27) 


54 


INTRODUCTION. 


where  F  y  stands  for  the  resistance  of  a  length  of  the  slide 
wire  equal  to  the  distance  of  y  from  the  center.  The  two 
parallel  wires  and  cross-piece  of  the  same  material  eliminate 
all  disturbance  from  thermoelectric  effects. 

69.  Measurement    of    Electrolytic    Resistance. — A    steady 
current  from  a  battery  and  a  galvanometer  cannot  satisfac- 


FIG.   17. 


torily  be  used  in  measuring  the  resistance  of  an  electrolyte,  for 
a  steady  current  produces  in  a  short  time  polarization  at  the 
electrodes.  This  polarization  leads  to  too  high  an  estimate 
of  the  resistance  of  the  electrolyte,  for  when  no  current  flows 
through  the  galvanometer,  the  three  other  arms  of  the  bridge 
are  balancing  the  potential  difference  necessary  to  overcome 


vf«i*u»r\wi   i    i 


RESISTANCE    OF   ELECTROLYTES.  55 

the  true  resistance  of  the  electrolyte  plus  the  potential  differ- 
ence required  for  overcoming  the  polarization  potential  dif- 
ference at  the  electrodes.  This  difficulty  is  obviated  by  using 
the  rapidly  alternating  current  from  the  secondary  of  an  induc- 
tion coil  instead  of  a  steady  current  from  a  battery  (Fig.  17). 
The  time  that  the  current  continues  in  one  direction  is  so 
short  that  no  appreciable  accumulation  can  form  at  the 
electrodes  to  produce  an  opposing  difference  of  potential. 

An  ordinary  galvanometer  would  not  be  affected  by  an 
alternating  current,  but  a  telephone  receiver  is  substituted, 
which  is  a  very  delicate  detector  of  an  alternating  current 
by  the  sound  emitted  from  the  diaphragm  set  in  vibration 
by  the  alternate  strengthening  and  weakening  of  the  magnet 
about  which  the  alternating  current  flows. 

The  best  type  of  induction  coil  is  one  with  a  very  light 
interrupter  so  that  the  number  of  the  alternations  is  very  high.1 

The  telephone  receiver  should  be  as  sensitive  as  possible. 
It  is  impossible  to  obtain  a  balance  for  which  there  is  no 
sound,  for  even  though  there  were  a  balance  for  a  steady 
current,  there  would  not  in  general  be  a  balance  for  varying 
currents  such  as  are  used  in  this  experiment,  owing  to  the 
inductive  electromotive  forces  of  capacity  and  self-induction 
in  the  resistance  coils.  If  a  bridge  box  is  used  and  there 
is  uncertainty  as  to  whether  a  small  resistance  should  be  added 
or  cut  out,  the  ear  is  often  assisted  by  adding  and  cutting  out 
a  larger  resistance  about  which  there  is  no  doubt.  On  com- 
paring the  change  of  tone  on  a  variation  of  this  latter  resistance 
with  the  variation  of  tone  with  the  uncertain  resistance,  one 
can  often  decide  whether  the  small  resistance  should  be  added 
or  not. 

70.  Conductivity  Vessels.  —  Figs.  18  and  19  represent  two 
satisfactory  forms  of  conductivity  vessels.  The  Arrhenius 
cell  (Fig.  1  8)  is  excellent  for  all  electrolytic  work,  except  for 
liquids  of  high  conductivity.  The  distance  between  the 

1  Gebriider  Ruhstrat,  of  Gottingen,  Germany,  manufacture  a  very  satisfactory 
induction  coil.  The  interrupter  is  surrounded  by  a  padded  box  so  that  no  sound 
escapes.  It  is  operated  by  one  dry  cell. 


INTRODUCTION. 


electrodes  of  the  other  type  (Fig.  19)  can  be  varied  and  it  is 
therefore  better  adapted  than  the  other  to  such  liquids. 

71.  The  platinum  electrodes  must  be  platinized;  that  is, 
covered  with  platinum  black.  Dissolve  some  platinum  scraps 
in  hot  aqua  regia.  When  cool,  dilute,  and  add  a  very  little 
lead  acetate.  The  exact  composition  is 
not  essential.  The  cells  should  be  filled 
with  this  solution  and  a  direct  current  at 
about  5  volts  should  be  sent  through 
the  cell  for  about  5  minutes.  The  cur-  [UU 


FIG.  18. 


FIG.  19. 


rent  should  then  be  reversed  for  about  5  minutes,  which 
should  be  followed  by  about  5  minutes  with  the  current  in  the 
original  direction,  and  these  reversals  should  be  continued 
until  the  electrodes  are  covered  with  a  black,  velvety 
deposit.  The  cell  should  be  carefully  washed  out  with 
distilled  water,  and  then  filled  with  dilute  sodium  hydroxide 
solution  and  a  weak  current  sent  through  in  each  direction 


RESISTANCE    OF    ELECTROLYTES.  57 

for  a  few  minutes,  to  remove  all  traces-  of  chlorine.  After 
several  more  washings  with  distilled  water,  the  cell  is  ready 
for  calibration. 

72.  Cell  Constant.  —  If  R  =  resistance  of  a  solution,  k  = 
specific  conductivity,  /=  distance  between  electrodes,  and 
A  =  equivalent  area  of  electrodes  (Eq.  7). 


TABLE  X. 

Specific  Conductivity  of  .02  Normal  Potassium  Chloride  Solution. 

(1.492  gr.  KC1  per  liter  of  solution.)! 


Temp. 

Sp.  Cond. 

10° 

I.996X  10-3 

18° 

2.399X10-3 

25° 

2.768X10-3 

C  is  a  constant  for  the  cell  and  is  determined  by  finding, 
with  great  care,  the  resistance,  R,  of  a  solution  of  known 
specific  conductivity,  k.  Then 

C  =  kR  (29) 

A  .02  n  solution  of  potassium  chloride  is  very  suitable  for 
calibrating  most  cells  (Table  X).  The  cell  should  be  in  a 
thermostat  (§81)  in  which  the  temperature  has  become 
constant,  for  the  conductivity  changes  greatly  with  change 
of  temperature. 

Measurement  of  Electromotive  Force. 

73.  Compensation  or  Potentiometer  Method. — An  unknown 
electromotive  force  is  usually  measured  by  comparison  with  a 
known.  The  best  method  of  comparison  is  the  compensation 
or  potentiometer  method,  illustrated  in  Fig..  20.  One  or  more 

1  Findlay,  "Practical  Physical  Chemistry,"  p.  160. 


INTRODUCTION. 


cells  (top  Fig.  20)  whose  e.  m.  f.,  E,  is  known  or  constant,  are 
connected  to  a  high  resistance,  ABC.  The  cell  of  unknown 
e.  m.  f.,  is  connected  through  a  galvanometer  and  key  to  two 
points,  A  and  B,  such  that,  on  closing  the  key,  the  galvan- 


(a) 


FIG.  20. 


ometer  is  not  deflected.  When  such  is  the  case,  the  potentials 
at  B  and  D  must  be  the  same,  hence  the  fall  of  potential  from 
A  to  B  (=i  r)  must  equal  that  from  A  to  D  (=X).  E=i  R' 
where  R'  is  the  resistance  of  the  entire  circuit. 


.X 


r  =  -^r. 


(30) 


If  the  resistance  of  the  battery  and  connecting  wires  is  neg- 
ligible compared  with  A  C,  we  may  let  R=R'  and 


f 

R 


(31) 


If  an  accuracy  of  half  of  one  per  cent,  is  sufficient  (which 
suffices  for  all  the  experiments  in  Chapter  IX)  and  E  consists 
of  Daniell  cells,  this  approximate  formula  may  be  used.  The 
electromotive  force,  E,  is  1.105  volts  (§339)  multiplied  by 
the  number  of  cells. 

If  a  higher  accuracy  is  required,  the  value  of  r  is  found, 
both  for  the  unknown   cell  X  and  for  a  Weston  or  Clark 


POTENTIAL    MEASUREMENTS.  59 

cell  (§§336-338).  If  the  total  resistance  Rr  is  constant  and  r 
is  the  resistance  A  B  for  the  cell  of  unknown  e.  m.  f.,  X,  and 
r'  is  the  corresponding  resistance  for  the  Weston  or  Clark  cell 
of  e.  m.  f.  E'. 

.     X=^E'  (32) 

Notice  that  the  like  poles  of  E  and  X  adjoin.  Do  not  seek 
necessarily  a  minimum  deflection  of  the  galvanometer,  since 
such  may  be  due  to  a  break  or  high  resistance  in  the  connec- 
tions. Vary  r  until  a  change  of  one  ohm  changes  the  direction 
of  the  galvanometer  deflections.  The  resistance  A  C  may 
consist  of  a  single,  high-resistance  box,  if  it  is  provided  with 
traveling  plugs,  by  which  connection  may  be  made  at  inter- 
mediate points,  such  as  B.  If  two  resistance  boxes  are  avail- 
able so  that  A  B  and  B  C  may  each  be  a  separate  box,  the 
resistance  R  may  be  kept  constant,  a  plug  being  inserted 
in  one  box  whenever  the  corresponding  one  is  withdrawn  from 
the  other.  Resistance  boxes  made  up  particularly  for  such 
potential  work  are  called  potentiometers.  They  are  more 
accurate  and  convenient  than  simple  boxes,  but  are  also  more 
complicated  and  expensive. 

74.  Capillary  Electrometer. — A  capillary  electrometer  may  be 
used  in  place  of  a  galvanometer.  The  theory  of  this  instru- 
ment is  given  in  §320.  Two  convenient  forms  are  illustrated 
in  Figs.  21  and  22.  Mercury  and  dilute  sulphuric  acid 
(i  :6  by  volume)  meet  in  a  capillary  tube.  If  the  two  liquids 
are  given  slightly  different  potentials,  the  meniscus  moves. 
If  the  potential  of  the  mercury  is  higher,  or  positive,  the 
mercury  rises  in  the  capillary  and  vice  versa. 

The  type  of  Fig.  22  is  convenient  because  the  nearly 
horizontal  capillary  requires  a  considerable  movement  of  the 
meniscus  for  a  small  change  of  level,  and  the  position  of  the 
meniscus  can  therefore  be  read  directly  on  a  scale.  The  other 
type  (Fig.  21)  is  more  easily  adjusted,  but  the  motion  of  the 
meniscus  is  much  less  and  should  be  followed  with  a  micro- 
scope. The  sensitiveness  of  both  is  about  .001  volt.  The 


6o 


INTRODUCTION. 


great  difficulty  with  capillary  electrometers  is  to  keep  them 
chemically  clean.  If  a  tube  becomes  foul,  it  is  better  to  dis- 
card it,  since  they  may  be  imported  from  any  German  dealer 
at  very  low  prices. 

Fig.  20  (a)  illustrates  how  the  capillary  electrometer 
may  be  used  in  place  of  a  galvanometer.  The  electrometer 
requires  a  key  consisting  of  a  strip  of  spring  metal  which 
ordinarily  makes  contact  with  a  point  above.  The  spring  is 
connected  with  one  electrode  of  the  electrometer  and  the 


FIG.  21. 


FIG.  22. 


upper  contact  is  connected  with  the  other  electrode,  so  that 
the  electrometer  is  normally  short-circuited.  Beneath  the 
spring  is  a  lower  contact.  The  two  points  between  which  the 
potential  is  to  be  determined  (Band Din  Fig.  20)  are  connected, 
one  to  this  lower  contact  and  the  other  to  the  upper  contact 
(which  is  also  connected  to  one  electrode  of  the  electrometer) . 
Therefore,  when  the  spring  is  depressed,  the  short  circuit 
is  broken  and  the  electrometer  is  connected  into  the  circuit 
(replacing  the  galvanometer  and  its  key  in  Fig.  20) . 

75.  Single  Potential  Differences,  Calomel  Electrode. — The 
difference  of  potential  between  a  metal  and  a  solution  cannot 
be  directly  determined  on  account  of  the  additional  difference 


ELECTRODE    POTENTIALS. 


6l 


of  potential  introduced  with  the  metal  by  which  connection  is 
made  to  the  solution.  We  therefore  require  a  metal  and  solu- 
tion whose  difference  of  potential  is  known  or  assumed. 

The  most  common  standard  single  difference  of  potential 
is  that  of  the  normal  or  tenth-normal  calomel  electrode.  This 
consists  of  mercury  in  contact  with  mercurous  chloride  crystals 
and  normal  or  tenth-normal  potassium  chloride  solution. 
The  mercury  is  usually  at  the  bottom  of  a  small  glass  bottle 
and  on  top  of  the  mercury  is  the  mercurous  chloride.  The 


FIG.  23. 

remainder  of  the  bottle  is  filled  with  the  potassium  chloride 
solution.  Through  a  rubber  stopper  pass  a  glass  tube  contain- 
ing a  platinum  wire  connecting  with  the  mercury  and  another 
glass  tube  which  is  filled  with  the  solution  and  which  serves 
as  a  syphon  to  connect  the  potassium  chloride  solution  with 
the  solution  to  be  tested.  If  the  solution  is  not  neutral  toward 
potassium  chloride,  they  should  be  connected  through  a  third 
solution  which  is  neutral  toward  both  (see  Fig.  23).  Both 
vessels  are  provided  with  a  third  tube,  with  which,  by  blowing 
in  or  sucking  out  air,  the  liquid  may  be  forced  into  the  syphons. 
The  electromotive  force  of  the  complex  cell  whose  electrodes 
are  the  electrode  being  tested  (A,  Fig.  23)  and  the  mercury 


62  INTRODUCTION. 

of  the  calomel  electrode  is  determined  by  connecting  it 
to  the  compensation  apparatus,  as  the  cell  X  is  connected  in 
Fig.  20.  The  potential  difference  of  the  normal  calomel 
electrode  at  temperature  /  is  usually  taken  as  (§318) 

E  =  . 560  [i+. 0006  (*-i8)]  (33) 

and  that  of  the  tenth-normal  electrode  as 

£  =  .613  [i  +.00008  (*-i8)]  (34) 

the  mercury  in  both  cases  being  at  the  higher  potential. 

76.  Signs. — After  the  potential  difference  has  been  deter- 
mined, great  care  is  required  in  determining  whether  the  above 
potentials  should  be  added  or  subtracted.  Remember  that  the 
positive  pole  of  the  complex  cell  is  the  pole  adjacent  to  the  positive 
pole  of  the  battery  at  the  top  of  Fig.  20,  and  that  the  mercury 
is  positive  toward  its  solution  by  the  above  values. 

For  example,  suppose  the  positive  pole  of  the  battery  of 
the  compensation  apparatus  is  the  one  marked  +  and  the 
galvanometer  or  capillary  electrometer  shows  a  balance  when 
the  resistance  between  A  and  B  =652  ohms  and  that  between 
B  and  C  is  4867  ohms.  We  will  suppose  further  that  the 
battery  consists  of  two  Daniell  cells  =  2. 21  volts. 

Hence  X  =--  —  2.21  =.262  volts 
5499 

If  X  consists  of  a  normal  calomel  electrode  and  a  metal 
electrode  with  intervening  solutions  (Fig.  23),  at  18°,  and  the 
mercury  is  next  A,  and  V  is  the  potential  of  the  metal  above 
its  solution 

.262  =.56  — V  /.  V  =  . 298  volts 

If  the  same  value  had  been  obtained  with  the  unknown  elec- 
trode next  A ,  we  should  have  had 

.262=F-.56  /.  ^  =  .822  volts 

for,  the  positive  mercury  electrode  is  adjacent  to  the  negative 
side  of  E  and  its  potential  must  be  overcome  by  V,  the  poten- 
tial difference  of  the  unknown  electrode. 


MEASUREMENT    OF    CURRENT.  63 

77.  Small  Known  E.  M.  F. — A  small  e.  m.  f.  whose  value  is 
accurately  known  is  often  required  for  calibration.  Fig.  24 
represents  a  convenient  arrangement  for  obtaining  such  an 
e.  m.  f.  A  high  resistance  R  (e.g.,  11,000  ohms)  is  connected 
to  a  Daniell  cell.  The  lead  wires  to  which  it  is  desired  to 


FIG.  24. 

apply  the  small  e.  m.  f.  are  connected  to  two  points  between 
which  the  resistance  is  r .  If  the  resistance  of  this  circuit  is 
very  high  compared  with  r,  the  e.  m.  f.  is 


(35) 


If  R  =  1 1000  and  r  =  10,  E  =  .0010045 


Measurement  of  Current. 

78.  Currents  above .  i  ampere  are  most  conveniently  measured 
by  a  good  ammeter,  such  as  a  Weston  or  Hoyt  (Whitney  Elec. 
Inst.  Co.)  instrument.  Smaller  currents  are  best  measured  by 
finding  the  fall  of  potential,  with  a  galvanometer  calibrated  as 
a  voltmeter  (§49),  when  the  current  passes  through  a  known 
resistance.  If  the  current  is  alternating,  an  electrostatic 
voltmeter,  or  an  idiostatically  connected,  Dolezalek  electrometer 
must  be  used.  If  a  direct  current  is  very  small,  it  may  be 
sent  directly  through  a  galvanometer  which  has  been  calibrated 
by  observing  the  deflection  by  a  current  from  a  small 
known  e.  m.  f.  (§77)  and  measuring  the  resistance  of  the 
circuit.  A  current  may  be  obtained  in  absolute  measure 


64 


INTRODUCTION. 


with  a  tangent  galvanometer,  but  the  magnetic  disturbances 
in  most  laboratories  render  the  instrument  so  impractical 
that  it  will  not  be  described. 

Measurement  of  Quantity  of  Electricity. 

79.  Copper  Voltameter. — Figs.  25  and  25  (a)  give  two  side 
views  of  a  convenient  type  of  copper  voltameter.  The 
anode  plates  A  are  clamped  against  a  brass  plate  B.  The 


B    : 


FIG.  25. 

cathode  plate  is  considerably  narrowed  above  the  solution 
and  passes  through  a  wide  and  long  slot,  and  is  clamped  to  the 
brass  strip  D  which  is  mounted  on  the  paraffined  wood  block  E. 
The  block  E  is  attached  by  screws  to  the  brasses  B  and  D. 
Connection  is  made  to  the  anode  plates  at  the  binding  point 
L  and  to  the  cathode  at  M.  The  ends  of  the  brass  B  rest  on 


COPPER    VOLTAMETER.  65 

the  rim  of  a  battery  jar  containing  the  solution.  A  solution 
containing  100  gr.  of  crystallized  copper  sulphate,  50  gr.  of 
sulphuric  acid  and  50  gr.-of  alcohol,  dissolved  in  a  litre  of 
distilled  water,  will  prove  satisfactory. 

Anode  and  cathode  plates  are  carefully  cleaned  with  sand- 
paper. The  cathode  is  then  washed  with  distilled  water  and 
rinsed  with  alcohol,  dried,  and  weighed  to  tenths  of  milli- 
grams by  the  method  of  oscillations  (§31).  The  cleaned 


I    B0    ! 


01 


FIG.  250. 

plate  must  be  handled  with  filter-paper.  Copper  will  not 
deposit  where  touched  by  the  fingers.  Mount  cathode  and 
anode  plates,  being  careful  that  the  cathode  plate  does  not 
touch  the  brass  B.  If  it  is  desired  to  know  the  current 
as  well  as  the  total  quantity,  note  the  exact  time  to  seconds 
at  which  the  current  commences  to  flow  and  the  exact  time 
5 


66  INTRODUCTION. 

when  the  current  ceases.  The  cathode  should  be  removed  as 
soon  as  the  current  ceases,  washed  with  distilled  water  followed 
by  alcohol,  dried  and  weighed  to  tenths  of  milligrams. 

If  m  is  the   gain  in  weight  in  T  seconds,  the  quantity  of 
electricity  in  coulombs  which  has  traversed  the  voltameter  is 

m  ,  AX 

(36) 


.0003292 
and  the  average  current  is 

•_  e 

Any  preliminary  adjustments  should  be  made  with  an'auxiliary 
cathode  in  place  of  the  one  which  has  been  carefully  cleaned 
and  weighed.  The  negative  wire  can  easily  be  ascertained  by 
attaching  the  other  wire  to  the  anode  plates  and  dipping  the 
supposed  negative  wire  in  the  solution.  If  it  is  immediately 
covered  by  a  heavy  dark  brown  deposit,  it  is  the  negative  wire 
and  should  be  connected  to  the  cathode.  The  current  in 
amperes  should  not  exceed  one-fiftieth  of  the  area  in  cm2  of 
both  sides  of  the  cathode  (i.e.,  a  current  density  of  .02). 

If  an  accuracy  higher  than  1/5%  is  required,  a  silver 
voltameter  should  be  employed.  This  voltameter  will  not  be 
described  here,  as  such  accuracy  is  rarely  attainable  in  electro- 
chemical experiments. x 

Measurement  of  Dielectric  Constant. 

80.  The  following  method  (De  Sauty's)  may  be  used  with 
high-resistance  liquids,  such  as  benzol,  kerosene,  etc.  The  con- 
nections, Fig.  26,  resemble  those  of  a  Wheatstone  bridge. 
/?!  and  R2  are  two  variable  resistances.  Cx  is  a  fixed  con- 
denser, for  example,  a  Leyden  jar  of  high  insulation.  C2  is  a 
variable  condenser,  illustrated  in  detail  in  Fig.  27.*  Two 
circular  brass  plates  A,  A',  about  ten  centimeters  in  diameter, 

1  Complete  directions  for  the  silver  voltameter  are  given  by  Guthe  in  Bui. 
Bu.  of  Standards,  1905,  I,  p.  349. 

2  Such  variable  condensers  are  supplied  by  Max  Kohl,  Chemnitz,  Germany. 


DIELECTRIC    CONSTANT. 


67 


are  mounted  on  brass  supports  B,  B.  One  of  these  is  fixed 
in  the  collar  D',  and  the  other  can  move  horizontally  through 
the  collar  D,  and  the  distance  apart  of  the  plates  can  be 
measured  by  means  of  a  scale  on  B  and  a  vernier  (§25)  on  D. 
D  and  Df  are  mounted  on  hard-rubber  pillars,  P  and  Pf .  A 


FIG.  26. 

glass  vessel  surrounds  the  plates  A  and  A'  with  the  liquid 
under  investigation. 

The  terminals  of  the  secondary  of  a  high-frequency  induc- 
tion coil  are  connected  to  the  points  L  and  M  (Fig.  26)  and  a 
very  sensitive  telephone  is  connected  to  N  and  P.  The  plates 
A  and  A'  are  separated  by  a  few  millimeters  of  air.  One  of  the 
resistances  is  made  several  thousand  ohms,  and  the  other  is 


68 


INTRODUCTION. 


adjusted  until  the  sound  in  the  telephone  is  a  minimum. 
When  such  is  the  case 


C~ 


(37) 


For,  if  no  current  flows  through  the  telephone,  Af  and  P  must  be 
at  the  same  potential.  Therefore,  the  fall  of  potential  in  each  con- 
denser must  be  the  same,  or 

e^   _  e2 
Cz      C2 

where   e^  and  e2  are  the  charges  on  the  plates.      But  et,  must  have 
flowed  through  RIt  and  e2  through  R2 

e2       Rt      C2 

Now  surround  the  plates  A  and  A '  with  the  liquid  and  deter- 
mine similarly  the  ratio  of  the  new  capacity  to  Cl.     The  die- 


B' 


FIG.  27. 

lectric  constant  is,  by  definition,  the  ratio  of  the  capacity 
with  the  liquid  to  that  with  air,  the  area  and  distance  apart 
of  the  plates  being  the  same.1 

A  less  accurate  method,  which  may  be  used  as  a  check, 

1  A  guard-ring  condenser  must  be  used  for  more  accurate  work.  For  partially 
conducting  liquids,  see  Nernst,  Zeit.  phys.  Chem.,  1894,  14,  p.  622;  Drude,  Zeit. 
phys.  Chem.,  1897,  23>  P-  267- 


THERMOSTATS.  69 

consists  in  keeping  the  resistances  the  same  for  the  liquid 
as  they  were  for  minimum  sound  with  air,  and  adjusting 
the  distance  apart  of  the  plates,  A  and  A',  until  the  sound 
is  a  minimum.  The  dielectric  constant  of  the  liquid  is  approxi- 
mately the  distance  apart  of  the  plates  in  the  liquid,  divided 
by  the  distance  apart  in  air. 

Since  the  capacity  of  two  parallel  plates  is  given  approximately  by 
the  formula 


where  A  is  the  area  of  either  plate,  d  is  the  distance  between  the  plates 
and  £  is  the  dielectric  constant,  if  d=C2 


Miscellaneous  Apparatus. 

81.  Thermostats.  —  Many  of  the  experiments  in  Chapters 
VI,  VII,  VIII,  and  IX,  require  a  constant  temperature. 
The  simplest  means  of  maintaining  apparatus  at  an  approxi- 
mately constant  temperature  is  to  place  it  in  a  bath  of  water 
which  is  protected  from  loss  or  gain  of  heat  by  wrappings  of 
asbestos,  felt,  etc.  The  water  should  constantly  be  stirred 
and  hot  or  cold  water  added  from  time  to  time,  as  required. 

It  is  much  more  satisfactory  to  maintain  the  temperatures 
constant  by  an  automatic  supply  of  heat  or  cold.  Fig.  28 
represents  a  toluene  thermo-regulator,  which  is  very  satis- 
factory for  temperatures  above  room  temperature.  A  bulb 
E,  about  15  cm.  long  and  about  3  cm.  in  diameter,  is  nearly 
filled  with  toluene,  and  is  placed  in  the  bath  whose  tempera- 
ture is  to  be  kept  constant.  The  rest  of  the  bulb  E,  the 
vertical  tube  F,  and  a  small  part  of  the  larger  tube  C  are  filled 
with  mercury. 

The  amount  of  mercury  is  so  chosen  that  the  inner  tube  at 
C  is  just  open  when  the  bulb  E  is  at  the  desired  temperature. 
The  gas  enters  at  A  and  goes  to  the  burner  at  D.  The  by-pass 
B,  is  controlled  by  a  screw  pinch-cock  which  allows  slightly 
less  gas  to  pass  than  is  required  to  maintain  the  desired  tern- 


7o 


INTRODUCTION. 


perature.  Toluene  is  a  very  expansive  liquid  and  slight 
fluctuations  in  the  temperature  of  the  bath  cause  the  inner 
tube  at  C  to  be  opened  or  closed. 

The  bulb  E  may  be  filled  by  removing  the  tube  A,  closing 
the  upper  part  of  C  by  a  cork,  connecting  B  to  an  exhaust 


FIG.  28. 


FIG.  29. 


pump  or  aspirator,  and  connecting  D  to  a  supply  of  toluene 
through  a  glass  tube  with  a  glass  stop-cock  which  is  attached 
to  D  by  a  short  length  of  rubber  tubing.  (Toluene  attacks 
rubber.)  The  cock  at  D  is  closed,  and  that  at  B  is  opened 
and  the  bulb  is  exhausted.  Then  B  is  closed  and  D  opened 
and  the  toluene  rushes  over  to  occupy  the  partially  exhausted 


THERMOSTATS.  71 

bulb.     It  may  be  necessary  to  repeat  the  operation  several 
times  and  to  place  E  in  warm  water. 

When  sufficient  toluene  has  entered,  pour  in  the  approxi- 
mate amount  of  mercury  required  and  re-exhaust.  The 
final  adjustment  of  the  mercury  had  better  be  done  with  a 
fine  pipette.  A  small  adjustment  may  be  made  by  raising 
and  lowering  A .  Air  bubbles  must  be  carefully  excluded. 


FIG.  30. 

Fig.  29  represents  a  less  sensitive  but  very  convenient 
gas  regulator.  The  gas  enters  at  A  and  passes  to  the  tube  D 
and  the  burner,  both  through  a  by-pass  at  B  and  also  through 
the  bottom  of  the  inner  tube  at  C,  if  the  bulb  E,  which  is  in 
the  bath,  is  below  the  desired  temperature.  A  fits  into  the 
vertical  tube  by  a  ground-glass  joint.  The  by-pass  B  consists 
of  a  small  hole  in  A  and  a  corresponding  slight  groove  in  the 
conical  ground-glass  neck  into  which  A  fits.  By  slightly 
turning  A,  the  opening  of  the  by-pass  may  be  regulated.  The 
amount  of  mercury  for  different  temperatures  is  adjusted  by 
the  piston  in  the  side  tube  F. 

Still  another  type  of  regulator  is  illustrated  in  Fig.   30. 


INTRODUCTION. 


FIG.  31. 


The  reservoir  and  cock  at  L  serve  to  regulate  the  amount  of 

liquid  in  the  bulb  E  and  tube  F.     The  final  adjustment  for 

a  particular  temperature  is  made  by  raising  or  lowering  A. 

A  10%  solution  of  calcium  chloride  is  a 

_  A  satisfactory,  liquid  for  E  and  F.     The 

/Y"  bottom  of  F  contains  mercury. 

If  it  is  desired  to  maintain  the  bath 
at  a  temperature  below  that  of  the  room, 
a  Foote  regulator  may  be  used.  Cold 
water  enters  by  the  tube  A,  Fig.  31,  and 
flows  out  through  the  tube  D  to  the 
bath,  unless  the  temperature  in  the  bath 
is  too  low,  in  which  case  the  mercury 
at  C  falls  and  the  water  leaves  by  the 
tube  B  to  the  waste.  C  represents  the 
mercury  column  of  any  one  of  the  regu- 
lators described  above. 

Unless  the  temperature  is  high  or  the  bath  is  large 
a  bunsen  burner  is  not  as  suitable  for  thermostats  as 
is  a  small  luminous  burner  provided  with  a  mica  or 
glass  chimney. 

82.  Stirrers. — A   good   stirrer   is   a   wheel,    with 
spokes  similar  to  propeller  blades,   mounted  hori- 
zontally,   near   the   bottom  of  the   bath,  with  the 
blades  so  turned  as  to  raise  the  water  as  they  are 
revolved  by  a  small  motor  outside.      A  light  fan 
similar  to  the  stirrer  but  with  larger  blades,  mounted 
on  the  top  of  the  vertical  axis  of  the  stirrer,  will 
usually  rotate  by  the  hot  air  rising  from  the  ther- 
mostat and  run  the  stirrer. 

The  Witt  stirrer  illustrated  in  Fig.  32  is  very  useful 
for  tubes  or  bottles.  The  liquid  enters  at  the  bottom 
and  is  centrifugally  thrown  out  at  the  sides. 

83.  Gas  Burette. — Gases  are  conveniently  collected 
in  a  Hempel  gas  burette  (Fig.  33).     The  gas  is  col- 
lected in   the  vertical  graduated  tube  afnd  the  reservoir  is 
adjusted  until  the  liquid  is  at  the  same  level  in  both.     The 


FIG.  32. 


MISCELLANEOUS    APPARATUS. 


73 


pressure  of  the  gas  in  the  graduated  tube  is  the  barometric 
pressure  less  the  vapor  pressure  of  the  liquid. 

84.  Gas  Condensing  Vessel. — Fig.  34  represents  a  useful 
vessel  for  condensing  such  liquids  as  nitrogen  peroxide.  The 
gas  enters  at  B  and  is  condensed  in  the  bulb  A  which  is  sur- 
rounded by  a  freezing  mixture.  The  neck  C  has  a  ground- 
glass  stopper.  When  sufficient  gas 
has  been  condensed,  B  is  closed,  D  is 
connected  to  the  apparatus  where  the 
gas  is  to  be  used,  and  A  is  warmed. 


FIG.  33. 


FIG.  34. 


85.  Gas  Seal. — It  is  sometimes  necessary  to  stir  a  liquid  in  a 
bottle  without  any  gas  escaping.  The  gas  seal  illustrated  in 
Fig.  35  will  be  found  quite  satisfactory.  The  bottle  must 
have  a  long,  wide,  uniform  neck  such  as  is  usually  the  case  with 
Bunsen  gas  washing  bottles.  The  stirrer  is  mounted  on  a  tube 
A  about  4  mm.  in  diameter.  This  is  surrounded  by  a  very 
slightly  larger  tube  B  which  is  firmly  held  by  an  expanded, 
single-hole,  rubber  stopper,  C.  A  short  piece  of  similar 
glass  tubing,  D,  is  fastened  to  A  by  rubber  tubing,  E.  The 


74 


INTRODUCTION. 


lower  edge  of  D  and  the  upper  edge  of  B  are  the  bearing  sur- 
faces and  are  as  smooth  as  possible,  and  are  lubricated  with 
vaseline.  An  outer  tube,  F,  has  a  diameter  intermediate  be- 
tween the  internal  diameter  of  the  neck  of  the  bottle  and  the 

external  diameter  of  B.  The  upper 
portion  of  F  is  contracted  and  at- 
tached to  A  by  rubber  tubing  G. 
The  space  above  C  is  filled  with 
mercury. 

86.  Drying  Tubes.— It  is    often 
necessary  to  fill  a  vessel  with  dry 
air.     The  apparatus  of  Ames  and 
Bliss,1  Fig.  36,  will  be  found  very 
convenient  for  this  purpose. 

By  means  of  the  three-way  cock, 
A,  (i)  a  steady  current  of  air  is 
drawn  through  the  sulphuric  acid 
and  calcium  chloride  vessels  and 
the  tubes  are  filled  with  dry  air, 
(2)  the  vessel  is  connected  to  the 
aspirator  and  exhausted,  (3)  the 
vessel  is  connected  to  the  drying 
vessels  and  is  allowed  to  fill  slowly 
with  dry  air.  The  operation  should 
be  repeated  several  times.  B  is  a 
plug  of  glass  wool  to  exclude  dust. 

87.  Purifying  Mercury. — Mechan- 
ical   impurities   may   be    removed 
from  mercury  by  filtering  through 
fine  holes  in  glazed  paper,  the  final 
portion    on    the    filter    being    dis- 
carded.    Amalgams  etc.,  are  best  removed  by  distilling. 

The  still  illustrated  in  Fig.  37  runs  continuously  with 
little  attention.  A  is  the  bulb  of  a  boiling  flask  and  is  about 
10  cm.  in  diameter.  It  is  sealed  to  a  glass  tube  C  about 
2  cm.  in  diameter.  The  distance  from  the  end  of  the  tube  to 

1  "Manual  of  Experiments  in  Physics,"  p.  484. 


FIG.  35. 


MISCELLANEOUS    APPARATUS. 


75 


the  top  of  A  should  be  about  10  cm.  greater  than  the  normal 
barometric  light.  The  bottom  of  the  reservoir  B  is  about  i  cm. 
below  the  end  of  C,  and  2  cm.  below  the  side  tube  D.  The 
height  of  this  reservoir  is  about  15  cm.  The  length  of  the 
inner  tube  from  the  top  of  the  bend  near  the  bottom  to  the 
side  tube  H  must  exceed  the  barometric  height.  The  upper 
end  of  L  is  about  i  cm.  below  the  top  of  A . 

After  the  still  has  been  set  up,  F  must  be  filled  with  pure 
mercury  and  E  with  the  mercury  to  be  distilled.     The  side 


FIG.  36. 

connection,  H,  is  connected  to  a  Geryk  or  other  vacuum 
pump  and  the  inner  tube  is  exhausted  until  the  mercury  rises 
within  a  few  centimeters  of  the  barometric  height  in  L  and  C. 
The  reservoir  E  is  adjusted  until  the  bulb  A  is  about  one-third 
filled  with  mercury.  The  cock  in  H  is  then  closed,  and  A 
is  heated  gently  by  the  ring  burner.  The  mercury  will 
continuously  distill  over  into  the  inner  tube  L  and  overflow 
into  the  reservoir  F. 


Calibrating  Measuring  Vessels. 

88.  The  true  volume  of  bulbs,  flasks,  graduates,  pipettes, 
etc.,  is  found  by  weighing  them  clean  and  dry,  and  then  full  of 
distilled  water  at  a  definite  temperature.  All  the  precautions 


76 


INTRODUCTION. 


in  weighing  mentioned  in  §§30-33  should  be  observed.     The 
volume  of  the  water  is  found  from  Table  LI.1 

89.  Burettes  may  be  calibrated  by  the  arrangement   (due 

to  Ostwald)  shown  in  Fig.  38. 
A  carefully  calibrated  pipette 
s  with  such  a  side  connection 

as  is  shown  in  the  figure  is 
connected  to  the  bottom  of 
the  burette,  care  being  taken 
to  exclude  all  air  bubbles. 
The  burette  is  filled  with 
water  at  the  temperature  for 
which  the  pipette  was  cali- 

J^H^  !L  brated.     The  pinch-cock,    or 

0      B      oB  glass  stop-cock,  at  S  is  opened 

JJJ^ME /^"PH  and    the    pipette    is    allowed 

IT 


E- 


FIG.  37. 


FIG.  38. 


to  fill  until  the  water  stands  at  the  zero  division  of  the  burette. 
Cock  C  is  then  opened  until  the  water  has  run  out  of  A  to  the 
lower  mark  D.  B  is  now  opened  and  A  is  filled  to  the  upper 

1  For  more  extensive  tables  of  water  volumes  and  general  directions  for 
calibrating,  see  Bui.  Bur.  of  Stand.,  1908,  4,  p.  553. 


MISCELLANEOUS    APPARATUS. 


77 


mark  E.  The  reading  on  the  burette  is  equal  to  the  corrected 
capacity  of  the  pipette.  A  is  emptied  to  D,  B  is  again  opened, 
and  A  is  filled  to  E  and  the  reading  for  twice  the  capacity  of  the 
pipette  is  determined,  etc.  The  results  of  such  a  calibration 
should  be  preserved  in  a  chart  or  a  table  of  corrections. 

Carbon-dioxide-free  Water  and  Alkali. 

90.  Liquids  which  must  be  preserved  from  contact  with  the 
carbon  dioxide  of  the  atmosphere  are  conveniently  kept  in 
a    reservoir    bottle   and   burette 

such  as  is  shown  in  Fig.  39 
(Ostwald).  A  and  A'  are  soda- 
lime  tubes.  The  burette  is  filled 
by  opening  B  and  sucking  the 
liquid  up  by  D. 

Vessels  and  liquids  may  be 
freed  from  carbon  dioxide  by 
drawing  through  them  a  current 
of  air  from  which  the  carbon 
dioxide  has  been  removed  by 
passage  through  soda-lime  tubes. 

Filling  and  Emptying  Bulbs,  etc. 

91.  To  fill  a  bulb  with  a  liquid, 
warm  with  the  hand  or  by  playing 
a  flame  about  some  distance  be- 
neath.     Remove  from  the  source 
of  heat  and  plunge  the  end  of  the 
stem  into  the  liquid.     As  the  air 
in  the  bulb  cools  liquid  will  be 
drawn  in. 

Never  allow  the  flame  for  an  FIG.  39. 

instant  to  remain  stationary  be- 
neath  the   bulb,    and   until   the   bulb   contains  considerable 
warm  liquid  do  not  allow  the  flame  to  touch  the  bulb,  and 
then  only  where  there  is  liquid.     Successively  warm  the  bulb 


;8  INTRODUCTION. 

and  allow  it  to  cool  a  little  until  the  bulb  is  filled.  When  nearly 
full  it  may  be  best  to  gently  boil  the  liquid  in  the  bulb.  When 
the  bulb  is  almost  full  the  liquid  can  be  made  to  expand  to 
fill  the  entire  stem.  Then  allow  it  to  cool  completely  while  it 
draws  over  liquid  from  the  beaker. 

To  expel  liquid,  warm  gently  with  the  bulb  so  turned  that 
the  stem  is  filled  with  the  liquid;  then,  invert  so  that  the  stem 
is  highest,  and  allow  to  partially  cool.  Repeat  until  all  the 
liquid  is  expelled. 

92.  Cleaning  Glass  Apparatus. — Glass  apparatus  can  usually 
be  satisfactorily  cleaned  with  hot  chromic  acid  followed  by 
repeated  washings  with  distilled  water  and  a  final  rinsing  with 
alcohol.     Grease  is  best  removed  with  caustic  potash  or  soda, 
followed  by  repeated  rinsings  with  water.     Steam  is  excellent 
for  a  final  purification  of  conductivity  vessels. 

93.  Cock  Lubricant. — An  excellent  lubricant  for  glass  stop- 
cocks which  must  be  air-tight  is  composed  of  equal  parts  of 

pure  rubber  gum,  vaseline  and  paraffine. 
The  two  latter  are  melted  together  and 
the  rubber  is  cut  in  small  pieces  and 
dissolved  in  the  heated  liquid. 

94.  Universal  Wax. — This  very  use- 
FlG'  4°*  ful   soft   wax   consists  of  one  part  by 

weight  of  Venetian  turpentine  and  four  parts  of  beeswax, 
which  are  thoroughly  mixed  and  worked  together.  The  ap- 
pearance is  improved  if  best  English  red  vermilion  is  added 
before  mixing.1 

95.  Clip  Connectors. — The  spring-clip  connectors  illustrated 
in  Fig.  40  are  very  convenient  and  satisfactory  for  all 
electrical  connections.2 

1  Ames,  and  Bliss,  p.  496. 

2  Made  by  Fahnestock  Elec.  Co.,  Brooklyn,  N.  Y. 


CHAPTER  I. 
GASES,  VAPORS  AND  LIQUIDS.1 

96.  Gases.  —  Following  the  custom  of  physicists  and  chemists 
since  the  time  of  the  Greek  philosophers,  we  shall  assume  that 
matter  is  composed  of  discrete,  indivisible  particles,  called 
molecules.     The  simplest  form  of  matter  is  the  gaseous  state, 
for  less  matter  is  present,  and  therefore  the  particles,  or  mole- 
cules of  matter,  being  further  apart,  are  more  independent 
and  obey  simpler  laws. 

The  variables  which  determine  the  state  of  a  gas  are  mass, 
m,  temperature,  t,  pressure,  p,  and  volume,  v.  These  variables 
are  not  independent  and  we  shall  determine  the  relations 
between  them. 

97.  Boyle's    Law.  —  Let    m    and    /    be    constant.     Robert 
Boyle,2  in  1  66  2,  discovered  as  a  result  of  careful  measurements 
that  under  these  conditions  the  volume  of   a  gas  varies  in- 
versely as  the  pressure,  or,  expressed  in  mathematical  language, 


.  •  .  pv  =  constant  (3  8) 

or,  if  /?j  and  vl  are  the  pressure  and  corresponding  volume  for 
one  state  and  p2  and  va  are  the  respective  quantities  in  a  dif- 
ferent state 

P1vl  =  p2v2  (38') 

If  p  is  the  density  of  the  gas  (§4) 

m 

„=_ 

1  General    references  for  Chapter  I:     Meyer,  "Kinetic  Theory  of  Gases," 
translated  by  Baynes;     Boynton,  "Kinetic  Theory";     Poynting  and  Thomson, 
"Properties  of  Matter,"  "Heat";     Winkelmann,  1905,  Volumes  I,  II,  and  III; 
"Stoichiometry,"  Young. 

2  "A  Defence  of  the  Doctrine  Touching  the  Spring  and  Weight  of  the  Air." 
London,  1662.     Also  in  "The  Laws  of  Gases,"  Barus. 

79 


80  GASES,    VAPORS   AND    LIQUIDS. 

Therefore,  other  expressions  for  Boyle's  law  are: 
—  =  constant 


(39) 


P,      P.  (39') 

98.  The  Kinetic  Theory  of  Gases.— We  shall  show  that 
Boyle's  law  is  a  direct  consequence  of  a  simple  theory 
respecting  the  nature  and  motion  of  the  molecules  of  a  gas. 
The  molecules  of  a  pure  gas  are  assumed  to  be  of  uniform 
mass  and  shape.  They  are  also  assumed  to  be  in  constant 


FIG.  41. 

motion,  and  when  they  collide  with  each  other  or  the  walls  of 
the  vessel,  the  coefficient  of  restitution  is  assumed  to  be  unity, 
or  no  energy  is  lost  by  the  impact.  These  three  assumptions 
are  in  accord  with  all  experience. 

Consider  a  body  of  gas  enclosed  in  a  rectangular  box,  the 
lengths  of  whose  edges  are  a,  5,  and  c  (Fig.  41) .     Let  m  =mass 


THE    KINETIC    THEORY    OF    GASES.  8  1 

of  a  single  molecule  of  the  gas.  Let  u  =  instantaneous  velocity 
of  the  molecule,  and  let  ua,  ub,  uc  be  the  components  of  this 
velocity  in  the  directions  of  the  three  edges  of  the  box.  If 
this  molecule  was  the  only  molecule  of  gas,  it  would  strike  the 
side  whose  area  is  be  with  a  momentum  mua  and,  rebounding 
with  an  equal  opposite  velocity,  would  give  this  side  a  total 
momentum  2  mua.  Before  it  again  strikes  this  wall  it  must 
travel  to  the  opposite  wall  and  back,  or  a  distance  2  a.  There- 
fore, the  number  of  times  which  it  strikes  the  area  be  in  one 
second  is 


2d 

and  the  total  momentum  per  second;  that  is,  the  force,  on  this 
wall  is 


a 

and  the  pressure,  or  force  per  unit  area,  is 

m  u 


abc 

Impacts  upon  other  molecules  and  the  other  faces  will  not 
affect  the  component  velocity  ua,  since  the  elasticity  is  perfect. 
If  there  are  N  molecules,  the  total  pressure  on  the  face  be  is 

p  =  ™^c  (ua*  +  ua"  +ua*"  +ua""  +  -  -) 

where  the  dashes  refer  to  the  different  individual  molecules. 
If  ua2  is  the  average  value  of  ua2  the  parenthesis  is  equal  to  N  ua2  . 

•••p=-^™^=™»»«*=(>^          (40) 

for  abc  is  the  volume  of  gas,  and  therefore 


__ 
abc 

is  the  number  of  molecules  per  cubic  centimeter  =n.     The 
number  per  cubic  centimeter  multiplied  by  the  mass  of  each 
6 


82 


GASES,    VAPORS   AND    LIQUIDS. 


molecule  is  the  total  mass  per  cubic  centimeter,  or  the  density 
p.  The  average  value  of  the  square  of  the  total  velocity  is 
given  by  the  equation 

w2  =  ua2  -f  ub2  +  uc2  =  3  ua2 

For  there  are  an  enormous  number  of  molecules  with  no 
preference  for  any  particular  direction  of  motion. 


/.  p  = — L  mnu2 

o 


(41) 


Therefore  if  u2  is  constant, 


is  also  constant.  We  shall  see  later  that  u2  depends  upon 
the  temperature,  and,  therefore,  if  the  temperature  is  constant, 
so  also  is 

—  and  pv. 

Therefore,  Boyle's  law  is  a  necessary  consequence  of  the 
kinetic  theory  of  gases. 


EXPERIMENT  I. 
Boyle's  Law.1 

A  50  c.c.  burette  and  a  similar  ungraduated  glass  tube  are  mounted 
on  adjustable  slides  on  each  side  of  a  vertical  scale  (meter  stick)  and 
are  connected  by  rubber  pressure  tubing.  Mercury  is  poured  in 
until  both  glass  tubes,  when  at  the  same  level,  are  about  half  full. 
The  burette  has  a  one-hole  rubber  stopper.  A  calcium  chloride 
drying  tube  is  inserted  in  the  stopper  and  the  burette  is  filled  with 
dry  air  by  lowering  the  other  tube.  Since  it  is  very  essential  that 
the  air  should  be  dry,  the  burette  should  be  filled  and  emptied 
several  times.  The  last  time  it  is  filled,  the  top  of  the  mercury 
column  should  be  left  in  the  middle  of  the  burette.  The  drying  tube 
should  now  be  replaced  by  a  tight  fitting,  square  end,  metal  rod 
which  is  inserted  until  it  is  flush  with  the  bottom  of  the  stopper. 
Observe  the  levels  of  the  two  mercury  columns  for  at  least  a  dozen 
carefully  measured  air  volumes  between  the  least  possible  volume 
and  the  greatest.  The  reading  of  the  scale  corresponding  to  the 
mercury  meniscus  may  be  found  with  the  help  of  a  square.  A 

1  The  apparatus  used  in  this  experiment  is  similar  to  that  devised  by  Duff 
and  described  in  his  "Elementary  Experimental  Mechanics."  (Macmillan.) 


VOLUMENOMETER. 


(a) 


piece  of  mirror  glass  behind  each  tube  obviates  error  from  parallax. 
Read  the  barometer  and  the  temperature. 

For  each  observation,  calculate  and  tabulate  the  volume  and  the 
pressure.  The  latter  evidently  equals  the  barometric  pressure, 
plus  or  minus  the  difference  of 
level  of  the  two  columns.  Plot 
the  results  in  a  diagram  in  which 
volumes  are  abscissae  and  pres- 
sures ordinates. 

QUESTIONS. 

1 .  Why  must  the  air  be  dry  ? 

2.  How  much  air  did  the  bu- 
rette   contain?       (Find    from  the 
curve    the     volume    at    76    cm. 
Calculate  the  density  from  Table 
LII  and  Equation  49.) 

3.  Calculate   the    volume    and 
density  if  the  pressure  was  (a)  i 
cm.  of  mercury,   (6)  200  cm.,  (c) 
10  dynes  per  sq.  cm. 

4.  (a)   What  is  the  constant  of 
equation  38  ?    (b)  Of  equation  39  ? 

99.   Volumenometer.  —  An 

interesting  practical  applica- 
tion of  Boyle's  law  is  the 
volumenometer  which  is  an 
instrument  for  measuring  the 
volume  of  bodies  by  finding 
the  volume  of  the  air  which 
they  displace. 

Replace  the  rod  in  the 
stopper  of  the  burette  in 
Fig.  42  by  a  glass  tube  which 
is  connected  by  pressure 
tubing  to  a  gas  washing 
bottle  (Fig.  420).  Let  P  = 
pressure  when  the 
tightly  closed  by 


(c) 


J 


rft 

I 

J 
J>, 

"  A 

i 

\j 

? 

/ 
/ 

! 

r 

C 

j 

^ 

1 

r 
i 

\ 

\ 

1 

/ 

FIG.  42. 


bottle  is 

a   rubber 

stopper  and  the  mercury  is  at  a  definite  division  A .  Lower 
the  mercury  to  a  point  B  near  the  bottom  of  the  burette, 
and  let  £=new  pressure.  Let  the  volume  of  the  bottle  and 
connections,  to  A ,  =  V  and  to  B  =  V  +v.  Then,  by  Boyle's  law, 

i  (42) 


84  GASES,    VAPORS   AND    LIQUIDS. 

Since  P,  p,  and  v  are  known,  V  may  be  calculated. 

Fill  the  bottle  about  half  full  with  a  carefully  weighed 
amount  of  an  assigned  salt,  sugar,  cement,  or  similar  body. 
Let  the  pressure,  when  the  mercury  is  at  A,  =P' ' ;  and  let  pf  be 
the  pressure  when  the  mercury  is  at  B.  The  unknown  volume, 
x,  of  the  substance  can  be  calculated  from  the  equation 

P'(V-x)  =p'(V-x+v)  (42') 

EXPERIMENT  II. 
Volumenometer. 

Clamp  the  burette  used  in  the  previous  experiment  and  connect 
a  150  c.c.  gas  washing  bottle  to  a  glass  tube  inserted  in  the  rubber 
stopper  (Fig.  42).  Use  heavy  pressure  tubing  and  make  all  joints 
tight  with  rubber  grease  (§93).  Close  the  bottle  with  a  tight 
stopper,  marking  how  far  it  goes  into  the  bottle.  Set  the  mercury 
at  a  division  A  near  the  top  of  the  burette  and  read  the  position  of 
each  mercury  column.  Lower  the  mercury  to  a  division  B  near 
the  bottom  of  the  burette  and  again  read  the  mercury  levels.  Repeat 
at  least  six  times,  returning  each  time  to  the  same  points  A  and  B. 

Weigh  carefully  about  75  c.c.  of  the  assigned  substance  and  pour 
it  into  the  bottle.  Make  six  more  observations  of  the  mercury 
levels  when  the  mercury  in  the  burette  is  successively  at  the  same 
two  divisions  A  and  B.  Observe  the  barometer  and  the  tem- 
perature. From  the  mean  positions  of  the  mercury,  calculate 
P,  p,  P',  and  p',  and,  finally,  x,  the  volume  of  the  substance,  and 
its  density. 

If  time  permit,  determine  the  density  also  with  a  pyknometer 
(§36).  Weighings  should  be  made  of  (i)  bottle  empty;  (2)  bottle 
filled  with  a  liquid  of  known  density  (Table  LIV)  which  is  inert 
toward  the  body;  and  (3)  containing  a  known  mass  of  the  body,  the 
rest  of  the  bottle  being  filled  with  the  liquid.  An  equation  for 
the  density  can  easily  be  worked  out. 

QUESTIONS. 

1.  What  would  be  the  most  suitable  values  of  v  and  x  for  a  given 
value  of  V?     (Estimate  accuracy-,  §20.) 

2.  Would  this  method  be  suitable  for  finding  the  density  of  a 
liquid?     Explain. 

3.  How  else  might  V  be  determined? 

4.  Calculate  the  weight,  in  kilograms,  of  one  cubic  metre  of  the 
substance  used. 

100.  Dalton's  Law. — In  1802  Dalton1  announced  the  law 
that  the  pressure  exerted  by  a  mixture  of  gases  is  equal  to  the 
sum  of  the  separate  pressures  which  each  gas  would  exert 

1  Mem.  Manch.  Lit.  &  Phil.  Soc.,  1802,  p.  535. 


GAY   LUSSAC'S    LAW.  85 

if  it  alone  occupied  the  space.  This  law  follows  immediately 
from  the  kinetic  theory.  For,  if  there  are  different  molecules 
of  masses,  ml}  m2,  etc.,  the  mean  squares  of  whose  velocities 
are  u^,  u22,  etc.,  the  total  pressure  on  the  walls  of  the  vessel  is 

«  III  222  jf I      I     f*      '  \T"O/ 

v5  O 

where  plt  p2,  etc.,  are  the  pressures  which  each  gas  alone  would 
exert.  An  illustration  of  Dalton's  law  is  given  in  Experiment 
VII. 

101.  Gay  Lussac's  Law.1 — This  law  states  that  the  propor- 
tional change  of  volume  of  a  gas,  for  a  given  change  of  tempera- 
ture under  constant  pressure,  is  the  same  for  all  gases  and 
numerically  equal  to  the  proportional  change  of  pressure  for 
the  same  change  of  temperature  when  the  volume  is  kept 
constant.  In  analytical  language,  if  vI  and  v0  are  the  volumes 
of  a  certain  mass  of  gas  at  the  temperatures  /T  and  t0,  the 
pressure  being  constant, 

VI—VQ  .  ,  . 

J ?~a»(*i— *J  (44) 

and  av  is  a  constant  which  is  the  same  for  all  gases.  Also, 
if  pT  and  p0  are  the  corresponding  values  of  the  pressure 
when  the  volume  is  kept  constant, 

Pl~Po=ap(tI-t0)  (45) 

and 

av  =  ap  =  a 

If  the  centigrade  scale  was  used  and  t=o,  Gay  Lussac 
found  that 

i 
"273 

1  Ann.  de  Chemie  et  de  Physique,  1802,  xliii,  p.  137.  Also  reprinted  (in 
original  French)  in  Mach,  "Principien  der  Warmelehre."  Charles  discovered 
this  law  several  years  before  Gay  Lussac,  but  did  not  publish  it. 


86  GASES,    VAPORS   AND    LIQUIDS. 


102.  Absolute  Temperature  and  Absolute  Zero.  —  Evidently 
if  tI  =  —  273,  pi  =o,  vt  =o,  and  therefore  this  temperature  was 
called  the  absolute  zero  and  temperatures  measured  from 
this  point  were  called  absolute  temperatures.  We  shall 
designate  the  absolute  temperature  by  0 

ft  ft 

(v  =  const.).     (48) 


o-  w 

Since  v  =  — 
i° 

°r^=^'  (/>  =  const.)  (49) 

103.  General  Gas  Equation.  —  We  shall  now  derive  an  equa- 
tion connecting  the  laws  of  Boyle  and  Gay  Lussac.  Suppose 
we  have  m  grams  of  gas  at  o°  occupying  a  volume  v0  under 
a  pressure  p0.  Let  us  increase  the  temperature  to  t°  at 
constant  pressure,  thereby  increasing  the  volume  to  vf  . 


273 

Now  change  the  pressure  from  p0  to  p  while  the  temperature  is 
kept  constant.  By  Boyle's  law  the  new  volume  v  will  be 
determined  by  the  equation 


V  p0  =  vp  =  p0  v0 

or 

pv  =  R'  0 
where 


ABSOLUTE    TEMPERATURE.  87 

R'  is  evidently  a  constant  for  a  given  mass  of  a  particular  gas, 
for  p0  is  any  arbitrarily  chosen  pressure  and  v0  is  the  volume 
of  this  gas  under  that  pressure  at  o°.  p0  is  usually  taken  as 
the  pressure  exerted  by  a  column  of  mercury  (density  =  13.59) 
76  cm.  high,  where  the  acceleration  of  gravity  is  980.6  dynes,  or 

p=  76  X  13.59  X  980.6  =1013200  dynes  per  sq.  cm.    (52) 
Dividing  Equation  50  by  m 

JL^P-Lor^^m*  (53) 

p        273^  P°e 

If,  for  example,  we  consider  i  gram  of  oxygen 

v=  -  — (Table  LII)  =699.8  c.c. 

.001429 

600.8  X  1013200 
/.  R'  =  -          —*^ =2-597X  io6 

/  O 

If  we  have  m  grams  of  oxygen 

pv=  2.597Xio6w#  (54) 

or 

^-  =  2.597Xio6# 
Similar  equations  may  be  derived  for  other  gases. 

104.  Absolute  Temperature  and  Kinetic  Energy  of  Mole- 
cules.— Comparing  this  equation  with  Equation  41,  we  see  that 
the  mean  square  of  the  velocity  of  the  molecules  is  propor- 
tional to  the  absolute  temperature.  Since  the  mass  of  the 
molecule  is  constant,  m/  2  times  the  mean  square  of  the  veloc- 
ity, or  the  mean  kinetic  energy  of  the  molecule  must  also 
be  proportional  to  the  absolute  temperature.  Moreover, 
Clausius1  demonstrated  that  in  any  mixture  of  gases,  the  col- 
lisions between  the  different  molecules  of  a  mixture,  reduce 
to  a  common  value  the  mean  kinetic  energy  of  each  kind  of 
molecule.  The  reader  is  referred  to  the  references  for  the 

i  Phil.  Mag.  1857  (4),  xiv,  p.  108.  Also  Maxwell,  Phil.  Mag.,  1860  (4), 
19,  p.  25,  Boynton,  Kinetic  Theory,  p.  39. 


88  GASES,    VAPORS    AND    LIQUIDS. 

proof  of  this  proposition.  Since  the  different  gases  of  the 
mixture  have  the  same  temperature,  equality  of  temperature 
must  mean  also  equality  of  the  mean  kinetic  energy,  J  win* 

PROBLEMS  I. 

1.  A  lo-litre  cylinder  contains  10  gr.  of  air  at  10°.      What  is  (a) 
the  density  of  the  air?      (b)  the  density  when  the  volume  has  been 
increased  to  i  5  litres  ?      (c)  the  pressure  in  each  case  ? 

2.  A  4o-litre  tank  is  filled  with  a  mass   of  oxygen  which  occupies 
10  litres  at  80  cm.  pressure,  nitrogen  which  occupies   50    litres  at 
40  cm.  pressure,  and  hydrogen  which  has  a  volume  of  20  litres  at 
100  cm.  pressure.      What  pressure  must  the  tank  sustain? 

3.  Calculate  the  mean  square  of  the  velocity  and  the  square  root 
of  the  mean  square  of  the  velocity  of  the  molecules  of  (a)  hydrogen, 
(6)  carbon  dioxide  at  o°  (Equation  41  and  Table  LII). 

4.  What  is  the  density  of  hydrogen  at  (a)   20°,  76  cm.  pressure? 
(Equation  49   and  Table   LII.)      (b)    at   20°  and   80   cm.   pressure? 
(Equation  53.) 

5.  A  loo-c.e.  glass  bulb  contains  i  gram  of  oxygen  at  30°;  what 
is  the  pressure  of  the  gas?     What  would  be  the  pressure  at  200°? 

6.  Calculate   the   gas   constant  R  for  one   gram  of  (a)  hydrogen, 
(b)  air. 

7.  What  is  the  mass  of  air  in  a  room  7  m.  x  5  m.  x  6  m.  at  20° 
and  70  cm.  pressure? 

8.  A  mass  of  gas  collected  over  mercury  at  25°  occupies  a  volume 
of  40  c.c.  when  the  mercury  is  10.5  cm.  above  the  general  level  out- 
side.    The   barometer  reads    74    cm.      What   is   the    volume   under 
standard  conditions  (76  cm.  and  o°)  ? 

9.  If  the  gas  were  nitrogen,  what  would  be  the  mass? 

EXPERIMENT  III. 
Coefficient  of  Increase  of  Pressure  of  Air. 

The  burette  of  the  apparatus  used  in  the  previous  experiments 
is  clamped  in  the  lower  half  of  the  slide.  A  bulb  of  about  100  c.c. 
capacity  with  a  capillary  stem  about  one  mm.  internal  diameter 
(Fig.  426)  is  filled  with  dry  air  by  exhausting  and  allowing  air  to 
re-enter  through  a  drying  tube  (§86).  The  bulb  should  be  ex- 
hausted and  filled  several  times  and  then  connected  by  pressure 
tubing  to  a  bent  capillary  tube,  the  other  end  of  which  is  inserted 
in  the  rubber  stopper  of  the  burette.  The  end  of  the  capillary 
should  be  flush  with  the  bottom  of  the  stopper. 

Surround  the  bulb  by  a  bath  of  ice  and  water.  The  mercury  in 
the  burette  should  be  brought  to  a  definite  mark  within  one  mm. 
of  the  stopper,  so  that  the  amount  of  air  outside  of  the  bulb  may  be 
a  minimum.  Observe  the  levels  of  the  two  mercury  columns. 
Readjust  the  mercury  columns,  and  reobserve  the  mercury  levels,  and 
repeat  at  least  five  times. 

Surround  the  bulb  by  water  at  about  10°,  adjust  the  mercury  in 
the  burette  to  the  former  mark,  and  again  observe  the  mercury 
levels.  Thus  find  the  difference  of  level  of  the  mercury  columns  for 
every  10°  until  the  water  boils,  at  which  temperature  at  least  five 


AVOGADRO'S    LAW.  89 

independent  observations  should  be  made.  Observe  the  barometer, 
the  temperature  of  the  room,  and  the  temperature  of  the  mercury 
in  the  manometer.  Calculate  the  pressure  for  each  temperature 
from  the  difference  of  level  of  the  mercury  columns  and  the  barometer 
height.  Both  should  be  reduced  to  o°  (§41).  Plot  the  pressures 
against  the  temperatures  as  abscissae.  The  points  should  lie  on  a 
straight  line.  Calculate  the  coefficient  of  apparent  increase  of 
pressure  from  the  pressures  for  o°  and  the  highest  temperature,  t° 
(approximately  100°),  since  at  these  temperatures  the  observations 
could  be  repeated  and  are  therefore  more  accurate.  The  apparent 
increase  of  pressure  is  the  observed  increase,  neglecting  the  ex- 
pansion of  the  bulb.  If  po  and  pt  are  the  observed  pressures,  the 
coefficient  of  apparent  increase  of  pressure  is 


If  /?  is  the  coefficient  of  cubical  expansion  of  the  material  of  the 
bulb 

vt=v0  (i+ftt) 

Where  vt  and  v0  are  the  volumes  at  t°  and  o°.  If  the  bulb  at  t°  had 
kept  its  original  volume  at  o°,  the  pressure  pt  would  have  been 
increased  to 


and  therefore  the  true  coefficient  of  increase  of  pressure  is 


pot 

o,PtP* 
pot   ^  p0t 

=  «'  +  /?  (55) 

for  the  second  term  is  so  very  small,  that  for  a  moderate  change  in 
temperature,  we  may  neglect  the  difference  between  pt  and  p0. 
The  value  of  ft  may  be  found  in  Table  LV. 

QUESTIONS. 

1.  A  litre   vessel  is  filled  with  air  at   20°  and   80   cm.   pressure. 
What  will  be  the  pressure  of  the  air  at  (a)  ioo°?      (6)  500°? 

2.  Why  must    (a)   the  air  be  dry?      (6)    a  capillary  connect  the 
bulb  and  the  manometer? 

3.  What  would  be  the  percentage  error  if  the  expansion  of  the 
bulb  was  neglected? 

105.  Avogadro's  Law.  —  In  1811  Avogadro1  announced 
the  law  that  there  are  the  same  number  of  molecules  in  two 
different  bodies  of  gas,  if  they  occupy  equal  volumes  at  the 

1  Jour,  de  Phys.,  1811,  Ixxiii,  p.  58. 


go  GASES,    VAPORS   AND    LIQUIDS. 

same  pressure  and  temperature.  This  law  of  Avogadro 
follows  at  once  from  the  kinetic  theory,  for,  if  we  distinguish 
the  different  quantities  for  the  two  gases  by  subscripts 


—  —  m 


If  we  divide  one  equation  by  the  other 


n2 


since  they  are  at  the  same  pressure.     (§98). 
since  the  temperatures  are  equal  (§104), 


Furthermore, 


106.  Molecular  Weights. — This  suggests  a  method  for 
comparing  the  masses  of  the  molecules  of  different  gases. 
Since  one  cubic  centimeter  of  each  gas  under  like  pressure 
and  temperature  contains  the  same  number  of  molecules, 
the  masses  of  each  kind  of  molecule  must  be  in  the  proportion 
of  the  masses  of  one  cubic  centimeter,  that  is,  the  densities. 
Oxygen  is  the  most  convenient  standard  gas  and  the  mass 
of  its  molecule  is  given  the  number  32.  If  p0  is  the  density 
of  oxygen  and  p  is  the  density  of  any  other  gas  at  the  same 
pressure  and  temperature,  the  molecular  weight  of  this 
gas  is 


If  both  gases  are  under  standard  conditions  (§4) 


M  = 


32 


.001429 


(56) 


(560 


107.  The  Molecular  Gas  Equation. — If  instead  of  using  the 
same  unit,  the  gram,  for  measuring  the  masses  of  different  gases, 
we  take  as  the  unit  of  mass  a  number  of  grams  numerically 
equal  to  the  molecular  weight  of  the  gas,  the  gas  is  said  to  be 
measured  in  gram-molecules.  Since  the  masses  of  equal 


MOLECULAR   GAS    EQUATION.  9  1 

volumes,  at  the  same  temperature  and  pressure,  are  propor- 
tional to  the  molecular  weights,  the  volume  of  a  gram-molecule 
is  the  same  for  all  gases. 

If  we  denote  by  w  the  mass  of  a  gas  expressed  not  in  grams, 
but  in  gram-molecules,  Equation  50  becomes 

pv  =  Rw6  (57) 

where  R  is  a  new  constant.     Since  equal  volumes  of  all  gases 
at  the  same  pressure  and  temperature  must  have  the  same 
value  of  w,  they  must  also  have  a  common  value  of  R. 
Since 

m  R'  M 

W  =  M>R  =  -^T  (58) 

and  since  we  have  already  found  that  R'  for  one  gram  of 
oxygen  is  2.597  Xio6,  R  must  be  8.31  Xio7  (ergs,  §108,  note). 

.'.  pv=  8.31  Xio^wd  (59) 

R  is  also  8.31  joules  =  1.985  calories  (§§5-7).        (59') 

If  p  is  expressed  in  centimeters  of  mercury,  rather  than  dynes 
per  square  centimeter 

pv=  6235  wO  (60) 

The  volume  of  one  gram-molecule  under  standard  conditions  is 
32  X699.8  =22.40  litres. 

EXPERIMENT  IV. 
Density  and  Molecular  Weight  of  Vapor  by  Dumas'  Method. 

A  glass  bulb  of  determined  volume  is  weighed  empty  and  when 
filled  with  vapor  at  an  observed  temperature  and  pressure.  From 
these  observations  is  deduced  the  density  under  standard  conditions. 
Let  m  =  original  weight  of  bulb,  mv  =  weight  of  bulb  filled  with 
vapor,  mw  =  weight  of  bulb  filled  with  water  at  t°,  t'  =  temperature 
of  bulb  when  sealed,  p  =  barometric  pressure,  pw  =  density  of  water 
at  t°  (Table  LI),  /?  =  coefficient  of  cubical  expansion  of  glass,  pa  = 
density  of  air  at  the  temperature  of  the  balance,  p'a  =  density  of  the 
air  at  *'  (Table  LII  and  Equation  53)  p'  =  density  of  vapor  (and 
possibly  some  air)  at  t1  .  The  volume  of  the  bulb  at  t'  is 

V=    * 


pw 


Q2  GASES,    VAPORS   AND    LIQUIDS. 

The  mass  of  the  vapor,  corrected   for   air   buoyancy,    is    (§33,    the 
correction  for  buoyancy  on  the  weights  is  negligible) 


mv  —  nt  +  Vp 

.*.*»'-          -r- 


A  bulb  which  has  been  cleansed  with  chromic  acid  and  distilled 
water  and  dried  with  alcohol,  is  weighed  to  milligrams  and  about 
one-tenth  filled  with  the  liquid  whose  vapor  density  is  to  be  deter- 
mined; for  example,  chloroform.  For  directions  for  filling  with 
liquid  and  emptying,  see  §91. 

The  bulb  is  suspended  in  a  large  bath  filled  with  a  liquid  which 
boils  at  a  higher  temperature  than  the  assigned  liquid,  and  is  heated 
until  all  the  liquid  in  the  bulb  has  evaporated.  The  neck  is  then 
sealed  with  a  small  flame  and  the  temperature  of  the  bath  noted. 
When  cool,  the  bulb  is  reweighed. 

The  bulb  is  filled  with  water  by  scratching  the  top  of  the  neck  with 
a  file  and  breaking  off  under  water.  Then  weigh,  with  the  broken 
tip.  If  a  bubble  of  air  remains  when  the  water  enters,  hold  the 
bulb  so  that  the  water  is  at  the  same  level  inside  and  out,  close  the 
neck,  remove,  dry  and  weigh.  Let  the  weight  be  m-w  •  Then 
completely  fill  the  bulb  with  water  (§91),  dry  and  weigh.  The 
weight  will  now  be  mw- 

Volume  of  air  = 


Pw 
The  volume  of  the  air  at  t'  is 


273 

If  p  is  the  true  density  at  t', 

p'  V=p'av+p  (V-v) 


Calculate  the  density  under  standard  conditions  (  §4)  .  Calculate 
the  molecular  weight  and  compare  it  with  that  represented  by  the 
formula. 

QUESTIONS. 

1.  What  corrections  have  been  neglected? 

2.  Why  must  the  liquid  entirely  evaporate? 

3.  Why  is  it  only  necessary  to  consider  the  buoyancy  of  air  on 
.the  vapor? 

4.  What  objection  would  there  be  to  sealing  the  bulb  at  a  tem- 
perature considerably  above  that  at  which  all  the  liquid  evaporates? 

5.  Calculate  the  weight  of  one  cubic  meter  of  this  vapor  at  20° 
and  76  cm.  pressure. 


DENSITY    OF   A    VAPOR. 


93 


EXPERIMENT  V. 
Density  and  Molecular  Weight  of  a  Vapor.     Victor  Meyer's  Method. 

A  mass  m  of  liquid  is  vaporized  and  displaces  a  mass  of  air  which 
occupies  a  volume  if  at  temperature  t  and  pressure  p.  The  volume  v 
under  standard  conditions  is  deduced  (§4),  and  the  density  is  found 
from  m  and  v.  The  liquid  assigned I  is  carefully  weighed  in  a  small 
glass-stoppered  bottle  and  introduced  into  a 
long  tube,  sealed  at  one  end  and  closed  by  a 
rubber  stopper  at  the  other  end  (Fig.  43).  This 
tube  is  surrounded  by  an  outer  tube  containing 
a  boiling  liquid1  of  much  higher  boiling  point. 
The  air  displaced  passes  through  a  side  tube  and 
is  collected  in  a  gas  burette  (§83).  The  inner 
tube  should  have  asbestos  or  glass  wool  at  the 
bottom  to  break  the  fall  of  the  small  glass  bottle. 
Be  careful  not  to  lose  the  small  bottles  or  mix 
up  their  stoppers.  Numbers  should  be  etched 
on  stoppers  and  bottles. 

Support  the  smaller  tube  inside  the  outer,  the 
bottoms  being  separated  by  several  cms.  Pro- 
tect the  glass  tubes  where  clamped  by  rubber 
tubing  or  asbestos  paper.  Pour  the  high  boiling 
liquid  into  the  outer  tube  to  a  depth  of  4  to  6 
cm.  The  outer  tube  must  be  well  protected 
from  the  flame  by  asbestos,  gauze,  or  a  sand- 
bath,  and  it  must  be  protected  from  drafts  by 
asbestos  wrappings. 

When  no  further  adjustment  of  the  reservoir 
of  the  gas  burette  is  necessary,  we  know  that 
the  pressure  and  therefore  the  temperature  have 
become  constant.  The  filled  and  carefully 
stoppered  and  weighed  glass  vial  is  placed  upon 
the  flexible  glass  detent  at  the  side  of  the  inner 
tube  and  the  rubber  stopper  is  replaced.  The 
reservoir  is  adjusted  until  the  water  in  it  and  the 
burette  are  at  the  same  level ;  and  if  the  pressure 
still  remains  constant,  the  burette  is  carefully 
read  and  the  bottle  is  released  from  the  detent. 
As  soon  as  the  bottle  reaches  the  heated  bottom 
of  the  tube,  the  liquid  will  escape  from  the  bottle 
as  vapor,  and  the  reservoir  must  be  lowered  to 
maintain  the  same  level  as  that  of  the  burette. 


FIG.  43. 


When  the  pressure  has  again  become  constant,  the  burette  is  care- 
fully read.  The  temperature  of  the  water  and  the  barometric 
height  should  be  ascertained.  The  difference  of  the  burette  readings 
is  v',  and  p  is  the  barometric  pressure  less  the  water  vapor  pressure 
(Table  LVII). 

Without  disturbing  the  apparatus,  two  or  more  other  bottles 
should  be  similarly  vaporized  and  the  equivalent  volumes  of  air 
determined. 

1  If  the  outer  boiling  tube  contains  water;  chloroform,  benzol,  ether,  or 
acetone  are  suitable  vaporizable  liquids.  If  a  less  volatile  liquid  than  these 
is  used,  the  water  in  the  outer  vessel  may  be  replaced  by  aniline  (boiling  point 
=  183°)  or  a  bromonaphthalene  (279°). 


94  GASES,    VAPORS   AND    LIQUIDS. 

Calculate  the  mean  density  of  the  vapor  at  o°  and  76  cm.  and  the 
molecular  weight  (Equation  56).  Compare  with  that  represented 
by  the  chemical  formula. 

For  the  determination  of  molecular  weight  by  this  method  at  high 
temperatures,  see  Nernst,  Zeit.  f.  Electrochemie,  1903,  p.  622. 

QUESTIONS. 

1.  Why  is  the  actual  temperature  of  the  vapor  unnecessary? 

2.  Should  the  air  displaced  by  the  vial  be  regarded?     Explain. 

3.  Calculate  the  weight  in  grams  of  one  cubic  meter  of  this  vapor 
at  (a)  o°  and  76  cm.;  (6)  20°  and  78  cm. 

EFFUSION. 

108.  A  gas  escapes  through  a  very  small  opening  with  an 
average  velocity  u.  The  kinetic  energy  of  a  mass  m  of  the 
escaping  gas  is  therefore  1/2  mu2  and  this  must  be  equal  to 
the  work  done  by  the  remaining  gas  or  pv.1  But  the  volume 
is  equal  to  the  mass  divided  by  the  density 

/,  p —  =  J  m  u2 


•4 


7  ,!.      <"> 

Therefore,  if  we  allow  two  different  gases  to  escape  through 
the  same  opening  under  the  same  difference  of  pressure,  the 
velocities  of  efflux  will  be  inversely  proportional  to  the  square 
roots  of  the  densities.  If  we  observe  the  times  7\  and  T2 
required  for  the  same  amount  of  gas  to  escape,  these  times  will 
be  inversely  proportional  to  the  velocities  and  therefore 
directly  proportional  to  the  square  roots  of  the  densities. 

•     P*  —      '  (6j'\ 

'  '  T~      T7  (      ) 

@2  -L    2 

PROBLEMS  II. 

1.  The  weight  of  a  certain  volume  of  air  is  5  grams.      What  would 
be  the  weight  of  a  similar  volume  (like  pressure  and  temperature) 
of  (a)  carbon  dioxide  ?      (b)  hydrogen  ? 

2.  What  is  the  pressure  of  a  gram-molecule  of  gas  which  occupies 
i  litre  at  o°? 

1  Work,  by  definition,  is  the  product  of  force  X  times  distance,  d,  but  if  the 
force  consists  of  a  pressure,  p,  applied  to  an  area  A ,  X  =  pA  and 

W=  pAd  =  pv  (62) 


COMPARISON    OF    DENSITIES    OF    GASES. 


95 


3.  Ten  grams  of  ammonia  gas  occupy  a  litre  vessel;  what  is  the 
pressure  at  (a)  o°?      (6)  ioo°? 

4.  The  density  of  a  vapor  is  .00497  at  20°  and  7°  cm-      Calculate 
its  molecular  weight. 

5.  What  is  the  volume  of  100  grams  of  nitrogen  at  30°  and  80  cm. 
pressure  ? 

6.  What  is  the  density  of  hydrogen  at  100°  and  TOO  cm.  pressure? 

7.  With  what  velocity  will  oxygen  escape  through  a  small  aperture 
at  25°  when  the  pressure  on  one  side  is  70  cm.  and  that  on  the  other 
side  is  74  cm.  ?     In  what  time  would  i  litre  escape  if  the  aperture  is 
.1  mm.  in  diameter? 

EXPERIMENT  VI. 
Comparison   of  Densities  of  Gases  by  Bunsen   Effusion   Apparatus. 

This  apparatus  consists  of  a  glass  tube  which  is  closed  by  a  three 
way  cock  and  which  is  filled  with  the  assigned  gas  and  depressed 
in  a  deep  mercury  reservoir  (see  Fig.  44).  A  float  rests  on  the 
mercury  inside  the  tube,  and  this  float  usually  has  a  black  knob  at 
the  top  and  two  black  lines  near  the  bottom.  One  position  of  the 
cock  connects  with  a  minute  aperture  in  a  thin  platinum  plate.1 
The  apparatus  should  stand  in  a  mercury  tray  so  that 
in  case  of  accident  no  mercury  may  be  lost.  The  side 
opening  of  the  three-way  cock  is  on  the  marked  side. 
With  the  interior  of  the  glass  tube  connected  with  the  air, 
lower  until  one  of  the  lower  marks  on  the  outside  of  the 
tube  is  on  a  level  with  the  mercury.  Close  the  cock  and 
lower  further  until  one  of  the  higher  marks  is  on  a  level 
with  the  mercury.  Connect  the  interior  of  the  tube  with 
the  fine  aperture  and  observe  with  a  telescope  the  number 
of  seconds  between  the  passage  of  the  lower  part  of  the 
black  knob  of  the  float  and  the  lower  of  the  two  lines 
near  its  bottom.  Repeat  several  times,  and  also  repeat  • 
with  the  tube  initially  depressed  to  a  different  mark. 
Make  similar  observations  with  several  other  gases. 

To  fill  the  tube,  raise  it  full  of  mercury.  Connect  the 
side  tube  to  the  source  of  gas,  and  opening  the  cock  a 
small  amount,  allow  the  tube  to  fill  as  the  mercury 
descends.  Fill  several  times  to  remove  any  residual  gas. 

From  the  mean  times,  calculate  the  relative  densities  of  the 
different  gases.  From  the  known  density  of  one  gas  (e.g.,  air  which 
under  standard  conditions  has  a  density  of  .001293)  calculate  the 
densities  of  the  others.  From  the  latter,  determine  the  molecular 
weights  (Equation  56)  and  compare  with  the  accepted  values. 

QUESTIONS. 

1.  Did  you  find  the  velocity  of  efflux  at  different  mean  pressures? 

2.  Should,  the  densities   be  reduced  to  standard  conditions  (§4)? 
Explain. 

3.  Why  must  the  aperture  be  small?     In  a  thin  plate? 

1  The  aperture  is  liable  to  become  clogged,  in  which  case  it  is  simpler  to  insert 
a  new  plate  than  to  clean  the  old  one.  A  hole  is  pricked  with  a  fine  needle  in 
thin  sheet  platinum,  and  then  hammfered  down  until  only  an  excessively  small 
hole  remains  in  a  flat  plate.  The  plate  is  trimmed  to  the  proper  size  and 
mounted  with  universal  wax  (§94). 


FIG.  44. 


96  GASES,    VAPORS   AND    LIQUIDS. 

Deviations   from  Boyle's  Law. 

109.  The  announcement  of  Boyle's  law  was  soon  followed 
by  the  discovery  of  exceptions.  The  most  careful  and 
exhaustive  study  of  these  exceptions  was  made  by  Amagat1 
whose  original  papers  must  be  consulted  for  details,  since  only 
the  briefest  possible  resume  is  possible  here.  He  carried  his 
observations  up  to  the  enormous  pressure  of  three  thousand 
atmospheres,  and  Witkowski,  Wroblewski,  Olszewski,  and 
Kamerlingh-Onnes  have  extended  the  study  of  common 
gases  to  extremely  low  temperatures. 

Table  XI  illustrates  the  deviations  of  three  typical  gases. 
Hydrogen  shows  comparatively  little  deviation,  while  the 
values  of  pv  for  carbon  dioxide  are  far  from  constant.  Air, 
nitrogen,  and  oxygen  show  greater  deviations  than  hydrogen. 
If  the  product  pv  is  taken  as  unity  at  one  atmosphere  and  the 
pressure  is  varied  at  different  constant  temperatures,  the  value 
of  this  product  is  a  minimum  at  some  particular  pressure  for 
each  temperature.  The  higher  the  temperature,  the  lower  the 
pressure  at  which  this  minimum  occurs .  For  air ;  this  minimum 
is  at  79  atmospheres  at  16°,  95  atmospheres  at  o°  and  123 
atmospheres  at  —78.5°.  Kamerlingh-Onnes2  found  no  per- 
ceptible minimum  with  hydrogen  until  very  low  tempera- 
tures were  reached.  At  —139.9°  the  minimum  was  at  less 
than  25  atmospheres,  at  —  195°  it  was  at  about  45  atmospheres, 
and  at  —213°  it  was  at  51  atmospheres. 

The  pressure  of  this  minimum  for  an  easily  condensed  gas, 
such  as  carbon  dioxide,  increases  as  the  temperature  is  raised 
till  a  certain  temperature  is  reached  beyond  which  it  behaves 
as  do  more  permanent*  gases.  At  o°  the  minimum  is  at  about 
4  atmospheres,  at  50°  at  about  130,  at  100°  at  about  200, 
at  200°  at  about  250,  but  at  250°  it  is  approximately  225 
atmospheres. 

1  Ann.  Chem.  et  Phys.,  1880,  19,  p.  345;   1893,  29,  p.  40;  29,  p.  37. 

2  Leyden  Communications,  1902,  78;  1907,  97,  p.  99. 

3  Gases  which  closely  follow  Boyle's  and  Gay  Lussac's  laws  under  ordinary 
conditions  and  which  remain  gases  to  very  low  temperatures  are  often  called 
permanent. 


VAN    DER    WAALS'    EQUATION. 


97 


TABLE  XI. 

Deviations  from  Boyle's  Law.1 
Values  of  pv  at  o°. 


Pressure 
(Atmospheres) 

i 

5° 

100 

200 

500 

1000 

2000 

3000 

Air                

i 

.068 

.072 

I.OIO 

1.340 

I.QQQ 

3.226 

4.323 

Hydrogen  

i 

1.032 

1.069 

1.138 

1.356 

!-725 

2.389 

Carbon  dioxide.  .  . 

i 

.105 

.202 

.385 

.890 

1.66 

no.  Van  der  Waals'  Equation. — Many  modifications  of 
Boyle's  law  have  been  proposed  for  the  purpose  of  meeting 
these  exceptions.  The  equation  which  on  the  whole  is  most 
satisfactory  was  invented  by  Van  der  Waals2  and  is 


(63) 


b  is  a  constant  which  represents  the  limiting  volume  of  the 
gas.  Obviously,  this  cannot  be  zero,  as  is  assumed  in  Boyle's 
law.  a  is  another  constant,  proportional  to  the  attraction 
between  the  molecules  of  the  gas.  Such  an  attraction  would 
evidently  increase  the  effective  pressure.  It  would  also  be 
proportional  to  the  square  of  the  density,  or  inversely  pro- 
portional to  the  square  of  the  volume,  for  the  closer  the  mole- 
cules are  together,  the  greater  the  number  of  both  attracting 
and  attracted  molecules. 

Deviations  from  Gay  Lussac's  Law. 

in.  Amagat3  also  made  a  classical  investigation  upon 
this  subject.  If  the  coefficient  of  increase  of  volume,  is 
measured  under  different  constant  pressures,  and  each  pressure 
is  greater  than  the  preceding,  the  value  of  this  coefficient 
generally  increases  to  a  maximum  and  then  decreases.  It 
varies  little  with  the  temperature. 

1  Amagat,  /.  c. 

2  "  Kontinuitat,"  1872  (Leipsic,  1881). 

3  Ann.  Chem.  et  Phys.,  1893,  29,  p.  68. 

7 


GASES,    VAPORS   AND    LIQUIDS. 


Table  XII  gives  typical  values  for  typical  gases.  It 
will  be  noticed  that  hydrogen  is  also  peculiar  in  this  respect. 
The  coefficient  of  increase  of  pressure  (constant  volume)  ex- 
hibits similar  deviations  as  Table  XIII  illustrates. 

TABLE  XII. 
Coefficient  of  Expansion  under  Constant  Pressure  (ay).1     o°-ioo°. 


Pressure 
(Atmospheres) 

i 

IOO 

200      * 

500 

IOOO 

Air 

00367 

004.4.4 

OO4.  tJ  tJ 

OO^  ^  I 

00214 

Hydrogen 

00366 

OO33  2 

.00278 

.00218 

Carbon  dioxide  

.00371 

.0414 

.OIII 

.00349 

.00206 

TABLE  XIII. 
Coefficient  of  Increase  of  Pressure  at  Constant  Volume 


o°-iooc 


Pressure' 
(Atmospheres) 

i 

50 

IOO 

200 

500 

Air  

00366 

00371 

' 
00462 

.00^2 

.00617 

Hydrogen  

.00367 

00373 

.00383 

.00379 

Carbon  dioxide  

.00369 

.00386 

•00373 

Comparing  Tables  XII  and  XIII,  we  notice  that  av  is,  at  ordinary 
pressures,  greater  than  ap,  except  in  the  case  of  hydrogen,  where  the 
reverse  is  true. 

Now,  using  the  notation  of  the  calculus 

_dp_          _  dv 
pdt  '  vdt 

Differentiating  Van  der  Waals'  equation  (63)  we  find 
R  R 


(64) 


<xv 


a      2ab 

pv — 

r        v       v* 


pv—pb 


If  a,  the  attraction  between  the  molecules,  is  negligible  compared 
with  b,  the  smallest  possible  volume  of  the  gas, 


R 


R 


Amagat  /.  c. 


ISOTHERMALS.  99 

Or,  in  the  case  of  hydrogen,  the  attraction  between  the  molecules 
must  be  much  less  than  in  the  case  of  the  other  gases.  If  b  is  negli- 
gible compared  with  a, 

R  R 

p  =  ~'''  av>ap 


pv  --- 


which  is  the  case  with  most  gases  at  moderate  pressures.     At  high 
pressures,  however,  all  gases  behave  like  hydrogen. 

The  explanation  of  these  deviations  is  the  .fact  that  when  a 
gas  is  cooled  and  sufficiently  compressed  it  changes  to  a  liquid, 
and  the  influence  of  this  transition  is  felt  at  pressures  and 
temperatures  considerably  removed  from  actual  liquefaction. 

Isothermals. 

112.  Fig.  45  is  constructed  from  Andrews'  and  Amagat's 
observations  upon  carbon  dioxide.1  The  line  marked  2i.°5, 
for  example,  shows  the  different  pressures  and  corresponding 
volumes  of  one  gram  of  carbon  dioxide  when  the  temperature 
is  maintained  constant  at  this  value.  Such  a  curve  is  called 
an  isothermal. 

At  low  pressures  (portion  A—  B  of  curve)  the  gas  approxi- 
mately obeys  Boyle's  law,  as  is  evident  from  the  shape  of 
the  curve.  When  the  pressure  reaches  about  60  atmospheres, 
the  volume  may  be  decreased  without  an  accompanying 
increase  of  pressure  until  the  volume  becomes  about  i  c.c. 
(B—C).  Upon  further  decrease  of  volume,  the  pressure 
increases  enormously  (C—D).  It  was  observed  that  at  B, 
liquid  commenced  to  appear,  and  as  the  volume  was  decreased, 
the  amount  of  liquid  increased,  until  at  C  all  the  carbon 
dioxide  had  become  liquid.  A  gas  which  may  thus  be  liquefied 
(states  A  —B}  is  called  a  vapor,  and  a  vapor  in  the  presence 
of  its  liquid  (states  B—C)  is  called  a  saturated  vapor.  Evi- 
dently, the  pressure  of  a  saturated  vapor  is  constant  if  the 
temperature  is  constant. 

Between  C  and  D  we  are  reducing  the  volume  of  the  liquid 
and,  since  liquids  are  very  incompressible,  the  pressure  greatly 
increases. 

1  /.  c.  and  "The  Laws  of  Gases."     Barus. 


100 


GASES,    VAPORS    AND    LIQUIDS. 


113.  Application    of    Van    der    Waals'    Equation. — Let    us 

apply  Van  der  Waals'  equation  (63)  to  one  gram  of  carbon 
dioxide   at    21. 5°  =  294. 5,    absolute,    taking   one   atmosphere 


100 


-80 


60 


40 


4567        89        10 

FIG.  45. 


as  our  unit  of  pressure,  and  the  cubic  centimeter  as  our  unit 
of  volume,  since  these  are  the  units  of  Fig.  45.  By  Equa- 
tion 51  and  Table  LIT 


R' 


273  X- 001965 


=  1.86 


PRESSURE    OF    SATURATED    VAPOR.  IOI 

For  one  gram  of  carbon  dioxide  a  =  i857,  6  =  .9  71  (Table  LVI, 
compiled   from   the   excellent   tables  in   Winkelmann,    1906, 

3,  P-  857). 


A  few  values  of  p  for  different  values  of  v  will  be  given. 
If  v  =  2o  c.c.,  £  =  24.2;  v  =  io,  p=42]  v  =  j,  £  =  52.7;  v  =  5, 
p=6i;  v=3,  £=66;  ^  =  1.5,  £  =  50;  v  =  i,  £  =  17000.  These 
points  are  represented  by  crosses  and  connected  by  a  curve. 
The  agreement  with  the  experimental  results  represented  by 
the  curve  A  BCD  for  the  same  temperature  is  not  exact,  but 
remarkably  close  considering  the  nature  of  the  curve.  The 
values  of  the  pressure  for  large  volumes  show  approximate 
agreement  with  Boyle's  law.  Between  8  c.c.  and  1.5  c.c.  the 
pressure  is  approximately  constant,  and  if  the  volume  is 
further  reduced  the  pressure  becomes  enormous. 

Prof.  James  Thomson x  made  the  suggestion  that  a  continu- 
ous curve  such  as  is  given  by  Van  der  Waals'  equation  repre- 
sents the  actual  physical  states  better  than  the  discontinuous 
curve,  with  the  horizontal  portion,  which  is  obtained  experi- 
mentally and  which  really  represents  the  mean  states  of  a 
mixture  of  liquid  and  vapor.  The  portions  BE  and  CF  can 
with  care  be  traced  experimentally  for  a  short  distance. 
The  portion  EF  represents  unrealizable  states,  since  a  decrease 
of  pressure  is  accompanied  by  a  decrease  of  volume. 

EXPERIMENT  VII. 
Pressure  of  Saturated  Vapor.     Dalton's  Law. 

The  same  apparatus  is  employed  as  was  used  in  Experiments  I, 
II,  and  III,  except  that  the  pressure  tubing  is  disconnected  from  the 
burette  and  is  connected  to  the  small  separating  funnel  shown  in 
Fig.  46.  The  funnel  must  have  a  tight  stop-cock.  It  may  be  neces- 
sary to  regrind  it  with  a  little  fine  emery  and  water.  It  should  be 
lubricated  with  rubber  grease  (§93).  The  funnel  is  surrounded  by 
an  open  glass  vessel  (see  figure)  containing  water  and  a  stirrer  and 
thermometer,  and  is  placed  directly  in  front  of,  and  as  close  as 
possible  to,  the  sliding  mirror  glass  behind  the  burette,  so  that  the 
level  of  the  mercury  can  be  read  as  in  the  previous  experiments. 

1  Proc.  Royal  Soc.  1871, .p.  130. 


102 


GASES,    VAPORS    AND    LIQUIDS. 


(A)  The  first  part  of  the  experiment,  although  not  more  im- 
portant, is  given  precedence  because  it  requires  that  the  burette  be 
perfectly  clean  and  dry,  in  which  condition  it  is  supposed  to  be  at 
the  beginning  of  the  experiment.  When  the  temperature  has 
become  constant,  introduce  sufficient  air  so  that  when  the  stop-cock 
is  tightly  closed  and  the  mercury  is  at  a  definite  point,  the  mercury 
in  the  other  arm  is  about  20  cm.  lower.  Make  several  careful 
independent  measurements  of  the  difference  of  level.  The  water- 
bath  must  be  stirred  constantly  and  the 
temperature  kept  constant.  If  there  is  no 
leak,  pour  a  little  of  the  assigned  volatile 
liquid  into  the  top  of  the  funnel,  lower  the 
other  mercury  tube  as  far  as  possible,  and 
open  the  cock  sufficiently  to  allow  a  very 
small  amount  of  liquid  to  enter  the  tube, 
and  close  tightly.  Adjust  the  mercury 
until  the  level  of  the  liquid  above  the 
mercury  is  at  the  former  point,  and  read 
the  difference  of  level  of  the  two  mercury 
columns  and  the  temperature.  There  must 
be  some  liquid  above  the  mercury  so  that 
the  vapor  may  certainly  be  saturated.  If 
there  is  much,  its  height  must  be  reduced 
to  mercury  (see  Table  LIV)  and  added  to 
that  of  the  mercury.  Make  several  care- 
ful independent  determinations  of  this 
difference  of  level.  The  temperature  must 
be  kept  constant.  Since  the  volume  of  the 
air  is  the  same  as  before  the  vapor  was 
admitted,  the  change  in  level  must  equal 
the  pressure  of  the  saturated  vapor,  if 
Dalton's  law  is  true  (§100). 

(B)  Remove  all  the  air  from  the  tube  by 
opening  the  stop-cock  and  elevating  the 
mercury  to  the  cock.  Close  the  cock  and 
lower  the  mercury  to  allow  any  air  left  in 
mercury  or  tube  to  come  out.  Again 
elevate  the  mercury  until  any  air  which 
has  collected  is  forced  to  the  top  and 
allowed  to  escape  through  the  cock  which 
is  then  closed.  Lower  the  mercury  and 
if  there  is  no  leak,  admit  a  little  liquid 
through  the  stop-cock.  The  temperature 
of  the  water-bath  must  be  the  same  as  in 
(A).  Find  the  difference  of  level  of  the 
two  columns  for  at  least  four  different  volumes  of  the  vapor.  Since 
the  pressure  of  a  saturated  vapor  is  independent  of  the  volume, 
these  differences  should  be  equal. 

Read  the  barometer  and  calculate  the  actual  mean  pressure 
exerted  by  the  vapor.  It  should  agree  with  that  determined  in 
(A),  thus  establishing  Dalton's  law.  Repeat  all  the  observations 
of  (B)  at  four  other  temperatures.  Tabulate  your  results  and  also 
plot  them  in  a  curve  with  temperatures  as  abscissae  and  pressures  as 
ordinates.  Another  method  of  measuring  the  vapor  pressure  is 
given  in  Experiment  IX. 


FIG.  46. 


HYGROMETRY.  103 

QUESTIONS. 

1.  What  was  (a)  the  pressure  exerted  by  the  air  in  (A)?      (6)  the 
total  final  pressure  ? 

2.  Why  must  (A)  be  performed  before  (B)  ? 

3.  What    difficulty    would    be   experienced    at   low   temperatures 

(§114)? 

4.  What  advantages  has  (A)  over  (£)  ?      What  disadvantages? 

5.  The  tube  of  the  funnel  will  generally  be  much  smaller  than  the 
other    sliding    tube.      Does    either    method    eliminate    error    from 
capillarity? 

114.  Hygrometry. — The   horizontal   lines  of  Fig.  45  which 
represent  the  coexistence  of  liquid  and  vapor  are  limited  by 
the  dotted  line*  LMN '.     If  the  pressure,  volume,  and  tempera- 
ture of  the  substance  correspond  to  points  on  the  left  of  this 
dotted  line,  the  substance  is  in  the  liquid  state;  if  the  points  are 
on  the  right  of  this  line,  it  is  in  the  vapor  state.     The  state  of 
the  water  vapor  in  the  atmosphere  would  be  represented  by 
the  latter  points  in  a  similar  diagram  for  water.     If  a  portion 
of  the  air  and  water  vapor  are  cooled,  the  pressure  will  remain 
equal  to  that  in  the  rest  of  the  atmosphere  and  the  different 
states  will  be  represented  by  points  on  a  line  similar  to  x  —y. 
The  temperature  corresponding  to  the  point  y  where  the  vapor 
becomes  saturated  and   moisture   appears  is  called  the  dew 
point.     The  dew  point  can  be  approximately  determined  by 
gradually  adding  cold  water  or  ice  to  water  in  a  brightly 
polished   metal   vessel,    stirring   constantly   and   noting   the 
temperature  of  the  water  when  dew  first  appears  outside  the 
vessel.     The  vessel  should  then  be  allowed  to  warm  up  while 
constantly  stirred  and  the  temperature  of  disappearance  of 
the  dew  observed.     The  mean  of  the  two  temperatures  is  the 
approximate  dew  point.     More  refined  methods  are  described 
in  §§42-44.     The  pressure  of  the  water  vapor  in  the  air  is 
evidently  the  saturated  vapor  pressure  for  the  temperature 
of  the  dew  point. 

The  relative  humidity  of  the  air  is  the  ratio  of  the  actual 
vapor  pressure  to  the  saturated  vapor  pressure  for  that 
temperature. 

115.  Critical  State,  Liquefaction  of  Gases. — Notice  that  the 
horizontal  lines  of  coexisting  liquid  and  vapor  (Fig.  45)  grow 


IO4  GASES,    VAPORS    AND    LIQUIDS. 

shorter  as  the  temperature  rises  and  disappear  for  isothermals 
above  30.9°.  Above  this  temperature  liquid  and  vapor  do  riot 
coexist,  and  therefore  it  is  called  the  critical  temperature. 
The  corresponding  pressure  is  called  the  critical  pressure.  The 
critical  volume  is  the  volume  occupied  by  one  gram  (or  one 
gram-molecule)  of  the  gas  at  this  temperature  and  pressure. 
The  characteristic  of  the  liquid  state  is  the  so-called  free 
surface;  that  is,  a  boundary  other  than  the  walls  of  the  con- 
taining vessel.  If  the  vessel  is  closed,  the  space  above  this 
free  surface  will  contain  the  saturated  vapor.  We  see,  there- 
fore, that  a  gas  above  the  critical  temperature  cannot  be 
liquefied.  It  may  be  compressed  to  a  smaller  volume  than 
the  liquefied  gas  occupies;  for  example,  in  the  state  represented 
by  the  point  P  in  Fig.  45,  the  carbon  dioxide  has  a  smaller 
volume  than  much  of  the  area  inclosed  by  the  dotted  line  or 
even  than  parts  of  the  area  at  the  left  of  this  line  but  no  free 
surface  will  appear.  To  liquefy  a  gas,  its  temperature  must  be 
lowered  below  the  critical  temperature,  and  the  volume  must 
be  decreased  by  increased  pressure  until  its  state  is  repre- 
sented by  a  point  at  the  left  of  the  right-hand  branch  of 
the  dotted  line.  The  methods  of  securing  the  necessary 
cooling  are  described  in  §§138-141. 

Van  der  Waals'  Generalized  Equation. 

116.  Corresponding  States. — If  we  multiply  out  Equation  63 
and  divide  by  p  we  have 

•+-^-|=0  -•;•  (6s) 

which  is  a  cubical  equation,  and,  therefore,  for  a  given  value 
of  p  there  are  either  three  real  values  of  v;  vlt  v2t  v3;  or,  one 
real  value  and  two  imaginary  values.  The  curve  in  Fig.  45 
plotted  from  the  numbers  in  §113  shows  these  three  real  roots 
for  certain  pressures;  for  example,  60  atmospheres.  Since 
the  three  real  roots  are  evidently  limited  to  the  area  included 
within  the  dotted  line  LMN,  the  higher  the  pressure  the  more 


CORRESPONDING    STATES.  105 

nearly  equal  they  must  be,  until,  at  the  critical  point,  they 
become  identical,  or 

vl  =  v2  =  v3  =  vc 

where  the  subscript  indicates  the  critical  state.     Equation  65 
must  be  equivalent  to 

(l)  —  7;.)  3  =  y3  —  7  y   yi  -4-  •?  y  2  y  —  qj  3  =  Q 
\^         "c/  o     c          '    O     c  c 

Equating  coefficients, 

3^=6+^1 

PC 

2_      a 
3=     ^ 

Hence,  dividing  the  third  equation  by  the  second, 

vc  =  3b  (66) 

Substituting  in  the  second  equation 


Finally,  substituting  in  the  first  equation 


•'•^=i-R  ^ 

and,  conversely, 

&=y  (68) 

a=3Pcvc'  (68') 


IO6  GASES,    VAPORS   AND    LIQUIDS. 

If  we  chose  as  our  units  of  pressure,  volume,  and  tempera- 
ture the  respective  values  for  the  critical  state,  and  if  we 
designate  these  so-called  reduced  values  of  the  pressure, 
volume,  and  temperature  by  X,  p,  and  v,  respectively, 

;=-|-,/-f  ,v  =  f.  (69) 

PC  vc  uc 

Substituting  these  values  of  p,  v,  0,  a,  b,  and  R  in  Van  rler 
Waals'  equation,  we  have,  upon  reducing 

i)  =  8V  (70) 

which  is  Van  der  Waals'  reduced  equation. 

Notice  that  it  involves  no  constants  peculiar  to  a  par- 
ticular substance.  In  other  words,  this  one  equation  states 
the  relation  between  the  reduced  pressure,  volume  and 
temperature  for  all  bodies  in  any  state  except  solid. 

117.  The  state  of  any  single  substance  is  completely  defined 
by  the  three  quantities,  pressure,  volume,  and  temperature, 
and  if  for  two  substances  the  reduced  values  of  these 
factors  are  equal,  the  substances  are  said  to  be  in  corre- 
sponding states. 

In  the  case  of  liquids  and  saturated  vapors  the  condi- 
tions are  somewhat  simpler,  since  the  volume  of  the  former 
is  practically  constant  and  the  pressure  of  the  latter  is  inde- 
pendent of  the  volume. 

As  an  example  of  the  application  of  this  generalized  equa- 
tion, we  will  consider  the  two  very  dissimilar  substances, 
sulphur  dioxide  and  ethyl  ether. 

For  sulphur  dioxide,  ^  =  78.9  atmospheres,  ^  =  428.4°. 
At  412.9°  absolute  the  pressure  of  the  saturated  vapor  of 
SO2  is  60  atmospheres. 

,       60  412.9 

.-.^=—-=.76;  v  =  -     ^  =  .96 
78.9  428.4 

For  ether,  ^=36.9;  ^  =  463.  If  p  is  the  actual  pressure 
in  atmospheres  of  ether  in  the  corresponding  state  to  that  of 


CORRESPONDING    STATES.  107 

the  SO2,  P  =  .  76X36.  9  =28.4.  Sajobschewski  found  that  sat- 
urated ether  vapor  had  this  pressure  at  445.8°  absolute. 
The  reduced  temperature  is, 


The  agreement  is  not,  however,  usually  as  close  as  this. 
If  the  experimental  values  of  p,  v,  and  0  are  substituted  in 
Equations  69  and  the  values  of  X,  //,  and  v  are  in  turn  substi- 
tuted in  Equation  70,  a  true  equation  is  not  obtained  unless  a 
variable  factor  is  substituted  for  the  number  8.  This  factor 
is  usually  different  for  the  gas  and  vapor  states  for  the  same 
A  and  v  (there  are  usually,  of  course,  three  values  of  /*  for  one 
value  of  A  and  v)  ,  but  the  mean  of  these  two  factors  is  usually 
approximately  8.  Moreover,  for  given  values  of  A  and  v, 
the  values  of  p  are  approximately  the  same  for  all  liquids, 
particularly  if  they  have  similar  constitutions,  and  /*  is  also 
approximately  the  same  for  their  vapors.  This  is  illustrated 
in  Table  XIV. 

PROBLEMS  III. 

1.  10  grams  of  air  are   contained  in  a   steel  cylinder  of  2  litres 
capacity,      (a)    What  is  the  pressure  at  20°?      (b)   To  what  volume 
must  it  be  compressed  to  increase  the  pressure  at  this  temperature 
to  1000  atmospheres? 

2.  A  certain  mass  of  hydrogen  occupies  i  c.c.  at  2000  atmospheres 
pressure.      What  would  be  its  volume  at  one  atmosphere? 

3.  Ten  grams  of  air  are  heated  from  20°  to  100°  under  a  constant 
pressure  of  100  atmospheres.      What  is  the  proportional  increase  of 
volume  ? 

4.  A  steel  vessel  of  one  litre  capacity  contains  air  under  a  pressure 
of  200  atmospheres  at  30°.     Calculate  the  pressure  upon  heating 
to  50°. 

5.  Calculate  the  pressure  for  typical  volumes  on  the  100°  isother- 
mal for  one  gram  of  ether  (a)  by  means  of  Van  der  Waals'  equation  ; 
the  constants  a  and  b  of  Table  LVI  and  R'  calculated  by  Equation  51. 
(6)   by  Van  der  Waals'  reduced  equation,  using  the  values  of  the 
critical    pressure,    critical    temperature   and    critical  volume    (  =  re- 
ciprocal of  critical  density),  given  in  Table  LVI. 

6.  Ten  grams  of  ammonia  gas  are  contained  in  a  litre  reservoir. 
Calculate  the  pressure  at  100°  by  Van  der  Waals'  equation,  taking 
a  and  b  from  Table  LVI,  and  calculating  R'  by  Equation  51.     pQ,  the 
reciprocal  of  v0,  may  be  calculated  from  the  molecular  weight  and 
Equation  56. 

'For  further  illustrations  see  Winkelmann,  1906,  3,  pp.  936-944. 


io8 


GASES,    VAPORS    AND    LIQUIDS. 


7.  The  dew  point  was  10°  when  the  temperature  was  22°.  What 
was  (a)  the  pressure  of  aqueous  vapor?  (6)  the  relative  humidity? 
(c)  the  amount  of  water  vapor  in  one  cubic  metre  of  air? 

TABLE   XIV.' 
Corresponding  States  under  the  Reduced  Pressure,  A  =  .08846. 


V 

H  liquid 

H  vapor 

Propyl  acetate 

7  <\4 

2QQ4 

2Q  7 

Methyl  acetate    . 

.7  ZOA 

.4006 

3O.  3 

Methyl  alcohol  

•7734 

•3949 

34-2 

Ethyl  alcohol 

7  704. 

4.O4  7 

•32    I 

Propyl  alcohol  

•7736 

.4028 

3I-I 

Acetic  acid  

.7624 

.4106 

*5-5 

Ethyl  ether  

•7371 

.4044 

28.2 

Benzol 

7282 

4O  C  3 

28   2 

Carbon  tetrachloride  .... 

•7251 

.4072 

27.4 

Stannic   chloride  

•7357 

.402  I 

28.1 

EXPERIMENT  VIII. 
Critical  State,   Isometric   and  Isothermal   Lines  of  Vapor. 

The  vapor  to  be  studied  is  contained  in  thick-walled  tubes  of 
about  3  mm.  internal  diameter,  and  4  cm.  long,  with  a  capillary 
stem  on  one  end,  through  which  they  are  filled,  and  which  is  after- 
ward sealed.  Each  contains  the  same  amount  of  ether  or  other 
substance2  which  should  occupy  about  one-fifth  of  the  tube.  The 
tubes  also  contain  different  amounts  of  mercury  to  vary  the  volume 
occupied  by  liquid  and  vapor.  The  least  volume  should  be  about 
one  and  a  half  times  the  volume  of  liquid,  and  the  greatest  volume 
should  be  about  four  times  the  volume  of  the  liquid.  At  least  four 
such  tubes  should  be  used  for  any  one  liquid. 

The  tubes  are  heated  successively  in  an  asbestos-lined  metal  box 
with  mica  sides  (Fig.  47).  The  junction  of  a  copper-constantin 
thermocouple  (§49)  is  secured  against  the  tube  by  a  wrapping  of 
thin  mica.  The  wires  pass  through  a  hole  with  asbestos  insulation 
and  are  connected  to  a  galvanometer  through  a  proper  resistance. 

1  From  Juptner,  I,  p.  49. 

2  Ether,  ethyl  alcohol,  benzol,  and  chloroform  are  suitable  liquids. 


CRITICAL    STATE. 


109 


The  deflection  of  the  galvanometer  at  disappearance  and  reappear- 
ance of  the  meniscus  is  carefully  observed  and  also  where  the  dis- 
appearance occurs,  whether  at  top  (all  liquid),  bottom  (all  vapor), 
or  intermediate  (critical  point.)  When  near  the  required  tem- 
perature, the  heating  or  cooling  should  be  very  gradual.  Estimate 
carefully  the  volume  occupied  by  liquid  and  vapor  in  each  tube, 


FIG.  47. 


which  is,  of  course,  the  volume  occupied  by  the  homogeneous  sub- 
stance above  the  observed  temperature.  Calibrate  the  galvan- 
ometer as  described  in  ^49  and  calculate  for  each  tube  the  tem- 
perature corresponding  to  the  mean  of  the  galvanometer  deflections 
at  disappearance  and  reappearance  of  the  meniscus. 

Before    plotting,    study    carefully    the    isothermal    diagrams    of 
Fig-  45- 


110  GASES,    VAPORS   AND    LIQUIDS. 

On  the  volume  axis,  lay  off  the  estimated  volumes  of  the  sub- 
stance in  the  different  tubes  and  at  each  point  erect  a  perpendicular 
(equal  volume,  or  isometric  line),  and  on  the  perpendicular  note  the 
observed  average  temperature  corresponding  to  this  volume,  and 
whether  the  homogeneous  substance  was  vapor  or  liquid. 

Mark  on  the  pressure  axis  the  critical  pressure  as  given  in  Table 
LVI.  Locate  the  point  on  the  critical  temperature  isothermal 
corresponding  to  this  pressure  and  the  observed  critical  volume. 
Draw  a  curve  through  this  point  similar  to  the  typical  critical  tem- 
perature isothermal  of  Fig.  45.  This  will  be  the  approximate 
critical  temperature  isothermal. 

Draw  a  short  horizontal  line  from  each  vertical  line  towards  in- 
creasing volume,  if  a  vertical  for  change  to  all  liquid,  and  vice  versa 
for  vapor  verticals.  Raise  or  lower  these  short  lines  until  they  are 
in  the  order  of  the  temperatures  and  until  their  junctions  with  the 
vertical  lines  lie  on  a  smooth  curve  tangent  to  the  critical  tem- 
perature isothermal  at  the  critical  -volume  and  pressure  (critical 
point).  Draw  this  curve  (curve  LMN  in  Fig.  45)  as  a  dotted 
line  and  prolong  each  horizontal  line  to  meet  it.  Complete  the  steep 
portions  (liquid)  and  the  hyperbolic  portions  (vapor),  and  write 
against  each  isothermal  its  temperature.  Since  the  exact  pres- 
sures were  not  measured,  the  positions  of  the  isothermals  will  be  only 
approximate,  but  the  form  of  the  curves  should  be  quite  exact. 

1 .  Estimate  the  critical  density. 

2.  What  would  be  the  influence  upon  the  observed  temperature 
of  air  in  the  tube? 

3.  Why  is  a  thermocouple  preferable  to  a  mercury  thermometer? 

Latent  Heat   of   Liquids. 

118.  A  change  of  state,  for  example,  from  vapor  to  liquid 
or  vice  versa,  involves  a  change  in  the  potential  energy  of  the 
molecules  and  possibly  also  external  work,  and  is,  therefore, 
always  accompanied  by  an  absorption  or  emission  of  heat 
energy.  This  heat  energy  per  gram  of  substance  is  called 
the  latent  heat. 

Although  the  latent  heats  of  fusion  and  vaporization  are 
accurately  determined  with  great  difficulty,  approximate 
experimental  methods  are  described  in  all  laboratory  manuals 
of  physics.  The  latent  heats  of  the  more  common  liquids  are 
given  in  Table  LVI. 

A  liquid  continues  to  give  off  vapor  from  the  surface,  or 
11  evaporate,"  as  long  as  the  pressure  of  the  vapor  above  the 
liquid  is  less  than  the  saturated  vapor  pressure,  independent 
of  the  total  atmospheric  pressure  above  the  liquid.  After 
the  pressure  of  the  vapor  reaches  the  saturated  vapor  pressure 


PRESSURE    OF    SATURATED    WATER    VAPOR. 


Ill 


for  that  temperature,  the  total  quantity  of  vapor  in  the  atmos- 
>here  above  the  liquid  remains  constant,  since  for  any  vapor 
liven  off  from  the  surface  an  equal  quantity  is  condensed. 
119.  Boiling. — If  heat  is  now  applied  to  the  liquid,   the 
miperature  will  not  rise  if   the  pressure    above  the  liquid 
imains  constant,  for  the  heat  absorbed  becomes  the  latent 
icat  of  the  increase  of  vapor.     If  the  total  pressure  above  the 
liquid  is  equal  to,  or  slightly  less  than  the  saturated  vapor 
pressure,  portions  of  the  liquid  below  the  surface  can  change  to 
vapor,  and  this  is  called  boiling.     If,  therefore,  the  pressure 
above  a  boiling  liquid  is  observed,  this  pressure  is  numerically 
equal  to  the  saturated  vapor  pressure  for  the  temperature  of 
the  liquid.     This  kinetic  method  of  determining  saturated 
vapor  pressure  (in  contrast  to  the  statical  method  of  Experi- 
ment VII)  was  invented  by  Regnault. 

EXPERIMENT  IX. 
Pressure  of  Saturated  Water  Vapor  by  Regnault's  Method. 

In  Regnault's  apparatus  the  total  pressure  above  the  surface 
of  the  liquid  can  be  kept  very  constant.  As  the  liquid  is  heated, 
the  vapor  is  condensed  in  a  Liebig  condenser  (see  Fig.  48),  and  as 
the  pressure  of  vapor  distributed  through  several  conducting  vessels 
is  the  vapor  pressure  cor- 
responding to  the  vessel  at 
lowest  temperature,  the 
pressure  exerted  by  the 
vapor  cannot  exceed  the 
maximum  pressure  corre- 
sponding to  the  tempera- 
ture of  the  tap-water  and  is 
therefore  very  small.  As 
the  temperature  of  the  boiler 
changes,  the  temperature  of 
the  air  in  the  boiler  varies, 
but  a  large  air  reservoir,  sur- 
rounded by  water,  is  con- 
nected between  the  con- 
denser and  the  manometer 
and  air-pump  or  aspirator, 
which  makes  the  volume  of 
the  air  in  the  boiler  small  FIG.  48. 

compared    with    the     total 

volume  of  air  in  the  system,  and  thus  the  increase  of  pressure  due  to 
the  heating  of  the  air  in  the  boiler  is  also  small. 

The  boiler  should  be  about  two-thirds  full  of  water;  fill  with  water 
the  small  tube  running  down  into  the  boiler  (which  tube  is  closed  at 
the  bottom),  and  insert  in  this  tube,  through  a  cork,  an  accurate 


112  GASES,    VAPORS    AND    LIQUIDS. 

thermometer.      Draw  out  any  water  which  may  be  in  the  air  reservoir 
by  means  of  a  stopper  in  the  bottom. 

Fill  the  surrounding  vessel  with  water.  Exhaust  the  air  from  the 
system  to  the  highest  vacuum  obtainable,  by  means  of  a  Geryk  pump 
or  aspirator.  Close  the  cock  through  which  connection  is  made  to 
the  aspirator  or  pump,  and  let  the  system  stand  a  few  minutes  to  see 
if  there  is  any  leakage.  If  not,  start  a  gentle  stream  of  water 
through  the  condenser,  and  place  a  Bunsen  flame  under  the  boiler. 
(Rubber  stoppers  will  hold  tighter  if  moistened  before  insertion.) 
Read  the  barometer  (see  §§40,  41). 

When  the  temperature  as  registered  by  the  thermometer  in  the 
boiler  becomes  very  steady,  we  know  that  the  water  is  boiling. 
Read  the  temperature  and  record  at  once  the  two  extremities  of 
the  mercury  column  of  the  manometer.  Let  in  a  little  air  by  slightly 
opening  the  cock  near  •he  air-pump  or  aspirator.  At  first  increase 
the  pressure  by  steps  of  about  i  5  mm.  Gradually  increase  the  steps, 
and  when  near  atmospheric  pressure  make  the  changes  of  pressure 
about  1 2  cm.  The  reason  for  the  difference  is  that  it  is  better  to  have 
the  steps  represent  about  equal  changes  of  temperature,  for  instance, 
about  5°.  Calculate  the  corrected  pressure  from  the  barometer 
reading  and  the  difference  of  level  of  the  mercury  columns.  By 
§119  this  pressure  is  equal  to  the  saturated  vapor  pressure  for  this 
temperature.  Tabulate  your  results  and  also  plot  them,  making 
temperatures  abscissae  and  pressures  ordinates. 

QUESTIONS. 

1.  State   precisely  what   you   have   observed  in  this  experiment 
and  what  relation  it  bears  to  saturated  vapor  pressure. 

2 .  What  condition  determines  whether  a  liquid  will  boil  or  evaporate 
at  a  given  temperature? 

3.  What  was  the  actual  pressure  of  water  vapor  above  the  boiling 
liquid?      (Table  LVII.) 

4.  What  determines  (a)  the  lowest  temperature,   (b)  the  highest 
temperature  for  which  this  experiment  is  applicable? 

1 20.  Mean  Vapor  Density.  Mathias'  Rule.1 — This  rule  states 
that  the  mean  of  the  densities  of  coexistant  liquid  and  vapor 
states  is  a  linear  function2  of  the  temperature.  If  pl  =  density 
of  liquid,  pv  =  density  of  vapor, 

ClO+c,        .,'.         ;"'".     .     (71) 

1  Jour,  de  Phys.,  1892,  3;  1893,  IX- 

2  One  quantity  is  said  to  be  a  linear  function  of  another  when  their  relation 
can  be  expressed  by  an  equation  which  involves  only  first  powers  and  constant 
terms  or  factors.     For  example,  y  is  a  linear  function  of  x  if  y  =  ax  +  b  where 
a  and  b  are  constants.     Such  a  relation  would  be  represented  by  a  straight  line. 
Also, 


dy 

~-  =  a  =  constant. 

doc 


MEAN    VAPOR   DENSITY. 


where  cx  and  c2  are  constants.  If  we  plot  density  against 
temperature,  the  density  of  the  liquid  will  decrease  if  the 
temperature  is  raised,  while  that  of  the  vapor  must  increase, 
until,  at  the  critical  temperature,  the  two  densities  become 
equal.  The  mean  of  the  two  densities  must  lie  upon  a  straight 
line,  inclined  to  the  horizontal  temperature  axis.  For  this 
reason  it  is  often  known  as  the  rule  of  the  straight  diameter 
(gerade  Mittellinie) .  The  experiments  of  Batschinski1  and 
others  have  shown  that  this  mean  line  may  be  regarded  as 
straight  with  little  error  in  most  cases.  At  the  absolute  zero 
we  may  reasonably  suppose  that  the  volume  is  a  minimum, 
and  this  would  require  by  Van  der  Waals'  equation  that 

v0  =  b  =  —    (Equation  68) 

Although  the  absolute  zero  has  never  been  attained,  exter- 
polation  from  experimental  results  indicates  that  the  volume 


Reduced  Temperature 
FIG.  49. 

at  the  absolute  zero  is  nearer  one-fourth  of  the  critical  volume 
than  one-third.  Therefore,  the  density  at  absolute  zero  is 
between  three  and  four  times  the  critical  density.  For 
reduced  temperatures  below  .7,  the  density  of  the  vapor  is 
negligible  compared  with  that  of  the  liquid.  Therefore  the 

1  Zeit.  phys.  Chem.,  41,  p.  741. 


114  GASES,    VAPORS    AND    LIQUIDS. 

mean  line  which  represents  the  average  of  the  densities  of 
vapor  and  liquid  and  which  passes  through  the  critical  density 
at  the  critical  temperature  must,  at  the  absolute  zero,  pass 
through  a  point  for  which  the  density  is  about  twice  the  critical 
density.  We  can  thus  construct  a  diagram  which  will  give 
an  approximate  representation  for  all  substances,  of  the  vari- 
ation in  density  of  liquid  and  vapor  as  the  temperature  is 
changed.  Such  a  diagram  is  shown  in  Fig.  49. 

121.  Since  below  the  reduced  temperature  .7  the  density  of 
the  vapor  is  negligible  compared  with  that  of  the  liquid, 
below  this  temperature  the  density  of  the  liquid  pt  is  repre- 
sented by  a  straight  line  and  is  therefore  a  linear  function  of 
the  temperature.  The  change  in  potential  energy  during 
change  of  state  is  evidently  dependent  upon  a/v2  the  term  in 
Van  der  Waals'  equation  which  expresses  the  equivalent 
pressure  contributed  by  the  attraction  between  the  molecules. 
The  internal  work  or  potential  energy  accompanying  evapor- 
ation is  therefore 

vv 


I' =   I    -—dv  =  a(  — — —  \  =ap 

J    v*  \vt       vv          '/ 


vi 

if  the  temperature  is  so  far  from  the  critical  temperature  that 
the  density  of  the  vapor  is  negligible  compared  with  that  of 
the  liquid.  Under  these  circumstances,  pt  is,  as  we  have  just 
seen,  a  lineal  function  of  the  temperature,  and  therefore  the 
internal  work  accompanying  evaporation  is  also  approxi- 
mately a  linear  function  of  the  temperature. 

SURFACE  TENSION. 

122.  This  attraction  between  the  molecules  also  manifests 
itself  as  surface  forces  and  surface  energy.  Molecules  near 
the  surface  are  subject  to  an  unbalanced  internal  force.  The 
dots  a,  b,  and  c  of  Fig.  50  represent  molecules  and  the  circles 
represent  the  sphere  within  which  lie  all  the  other  molecules 
which  exert  an  appreciable  attraction  upon  a,  b,  or  c.  The 


SURFACE    TENSION.  115 

shaded  portion  represents  the  molecules  which  exert  an  un- 
balanced attraction. 

This  resultant  inward  force  will  evidently  tend  to  make 
the  surface  as  small  as  possible  and  an  increase  of  surface 
requires  work  and  increases  the  energy  of  the  liquid.  The 
energy  per  unit  area  is  called  the  surface  tension. 

Since  work  or  energy  is  equal  to  force  times  distance,  the 
surface  tension  is  also  equal  to  the  tangential  force  across 
any  line  in  the  surface  of  one  centimeter  length. 


FIG.  50. 


123.  Eotvos'  Equation. — Since  surface  tension  depends  on 
the  same  physical  quantities  as  evaporation,  it  will  show  the 
same  variation  with  the  temperature.  Therefore  the  surface 
tension  is  a  linear  function  (§120,  note)  of  the  temperature  or 


dT 
dt 


(73) 


when  T  is  the  surface  tension   and   c'   is   a   constant.     The 
volume  of  a  sphere  containing  one  gram  molecule  of  liquid  is 

M 


where  M  is  the  molecular  weight  and  p  is  the  density.  The 
area  of  the  sphere  which  contains  one  gram  molecule  is 
therefore 

§ 


(73') 


Il6  GASES,    VAPORS   AND    LIQUIDS. 

c  should  have  the  same  value  in  every  liquid,  since  there  are 
the  same  number  of  molecules  in  a  mass  equal  to  the  molecular 
weight.  This  equation  was  discovered  by  Eotvos.1  While  c 
is  not  absolutely  constant,  it  is  quite  nearly  so,  and  has  the 
approximate  value  —2.1.  By  integrating  73  'between  tl  and  t2 


(74) 


TI  and  T2  and  pt  and  p2  are  the  values  of  the  surface  tension 
and  density,  respectively,  at  the  two  temperatures  t-,  and  ta. 
This  is  one  of  the  few  methods  available  for  determining 
the  molecular  weight  of  a  liquid.2 

124.  Measurement  of  Surface  Tension.  —  The  simplest 
method  of  measuring  the  surface  tension  of  a  liquid  is  to 
measure  the  height  to  which  a  liquid  will  rise  in  a  capillary 
tube.  We  shall  suppose  that  the  liquid  wets  the  tube  and 
that  the  liquid  has  been  raised  in  the  tube  so  that  there  is  a 
film  of  liquid  over  the  interior.  Let  r  be  the  radius  of  the 
tube  and  h  the  height  to  which  the  liquid  rises.3  If  T  is  the 
surface  tension,  the  force  upward  due  to  the  surface  tension  is 
27irT  and  the  downward  force  is  the  weight  of  the  column 
of  liquid,  or  nr2hpg  where  g  is  the  acceleration  of  gravity. 
At  equilibrium  the  two  must  be  equal 


(7S) 


1  Wied.  Ann.,  27,  p.  448. 

2  For  the  application  of  Eotvos'  equation  to  solutions,  see  Zemplen,  Ann.  d. 
Phys.,  1906,  20,  p.  783;  1907,  22,  p.  391;  and  for  the  application  to  fused  salts, 
see  Lorenz-Kaufler,  Ber.,  1908,  41,  p.  3727. 

3  If  the  liquid  wets  the  tube,  the  surface  in  the  tube  is  concave  and  the  height 
h  is  the  distance  from  the  general  outside  level  to  the  bottom  of  the  meniscus, 
plus  one-third  the  radius  of  the  tube.     For  the  volume  of   liquid  above  the 
meniscus  is 

TI  r  $  r 

7ir2r  —  ?>  nrz  = =  it  r2 

3  3 

If  the  liquid  does  not  wet  the  tube  approximately  §  must  be  subtracted  from 
the  height  of  the  meniscus. 


SURFACE    TENSION. 


117 


TABLE  XV. 

Surface  Tension  T  (15°),  Temperature  Coefficient  of  Surface 
Tension  c',  and  Angle  of  Contact  a. 


T 

c' 

a 

Ethyl  ether  

iQ 

•  —  .  I  I 

1  6° 

Ethyl  alcohol  

2  <C 

.087 

o° 

Benzol  

3  I 

•  I  1 

o° 

Water  

76 

.  I  <\ 

small 

Mercury  

527 

-•38 

i35° 

EXPERIMENT  X. 
Measurement  of  Surface  Tension.    Eotvos'  Law. 

Clean  the  assigned  capillary 
with  hot  chromic  acid,  dis- 
tilled water  and  alcohol,  and 
dry.  Pour  the  assigned  liquid 
into  a  75  c.c.  side-neck  test- 
tube1  to  a  depth  of  about  2 
cm.  Introduce  a  mirror-glass 
scale  and  insert  the  capillary 
through  a  rubber  stopper 
(Fig.  51).  Attach  a  rubber 
tube  to  the  side  connection. 
If  the  liquid  is  volatile  the 
tube  should  normally  be  closed 
by  a  spring  clamp.  Raise  and 
lower  the  liquid  in  the  capillary 
several  times  by  blowing  or 
sucking  with  the  rubber  tube. 
Read  the  top  of  the  meniscus 
and  the  under  level  of  the 
liquid  in  the  test-tube.  Par- 
allax is  avoided  by  reading 
the  position  in  the  mirror  of 
the  image  of  the  liquid  surface 
when  the  surface  itself  and  its 
image  appear  coincident. 

Repeat  the  readings  at  least 
thfee  times  with  the  liquid 
previously  elevated  in  the 
capillary  and  an  equal  num- 
ber with  the  liquid  previously 
depressed.  The  temperature 
of  the  water-bath  must  be 

1  Findlay,  Practical  Physics,  p.  78. 


FIG.  51. 


Il8  GASES,    VAPORS   AND    LIQUIDS. 

maintained  constant  within  0.2°.  Raise  the  temperature  of  the 
water-bath  about  20°.  When  the  temperature  has  become  constant, 
repeat  all  the  observations.  The  water  must  be  stirred  constantly 
and  the  temperature  maintained  constant  within  0.5°.  Remove 
the  capillary,  break  it  at  the  mean  position  of  the  meniscus,  and 
measure  two  diameters,  at  right  angles,  with  a  micrometer  micro- 
scope (§27).  Repeat  the  measurement  several  times  and  take  the 
mean.  The  whole  work  should  be  repeated  with  as  many  tubes  of 
different  sizes  as  time  will  permit.  Calculate  the  surface  tension 
at  both  temperatures  by  Equation  75  and  also  its  temperature 
coefficient  (Eq.  73).  Find  the  mean  density  of  the  liquid  from  Table 
LIV,  or  by  experiment,  and,  assuming  M  and  p  constant  for  this 
range  of  temperature,  calculate  the  value  of  M  from  the  equation 

(74'} 


QUESTIONS. 

i.  Are  the  errors  of  measurement  sufficient  to  explain  the  differ- 
ences between  results  with  different  tubes? 
2  .    What  other  sources  of  error  may  there  be  ? 

3.  How  could  the  surface  tension  of  mercury  be  obtained  in  an 
analogous  way? 

4.  How  high  would  the  liquid  used  rise  in  a  tube  o.i   mm.   in 
diameter  at  the  first  temperature? 


Viscosity. 

125.  Another  property  of  liquids  which  largely  depends  upon 
these  molecular  forces  is  the  opposition  to  relative  motion  of 
different  portions. 

A  solid  has  rigidity;  that  is,  it  offers  a  continued  resistance 
to  forces  tending  to  change  its  shape.  A  liquid  has  no 
rigidity  and  offers  no  continued  resistance  to  forces  tending 
to  change  its  shape;  that  is,  the  smallest  force  if  given  time 
will  produce  an  unlimited  change  in  the  shape  of  the  liquid. 
But  the  rate  at  which  a  liquid  changes  its  shape  under  a  given 
force  is  not  the  same  for  all  liquids.  Some  liquids  change 
very  slowly  and  are  called  viscous  liquids,  others  change 
rapidly  and  are  called  mobile  liquids.  The  action  of  both  can 
be  stated  in  terms  of  a  property  called  viscosity. 

126.  The  viscosity  of  a  fluid  may  be  denned  as  the  ratio  of 
the  shearing  stress  in  the  fluid  to  the  .rate  of  the  shear.     From 
this  general   definition   a   simpler  definition   can  be   readily 


VISCOSITY. 

deduced.  A  shear  consists  essentially  in  the  sliding  of  layer 
over  layer  and  the  shearing  stress  is  the  force  per  unit  area 
required  to  produce  the  shear.  Hence  we  have  the  following 
equivalent  definition:  "The  coefficient  of  viscosity  is  the 
tangential  force  per  unit  of  area  of  either  of  two  horizontal 
planes  at  unit  distance  apart,  one  of  which  is  fixed,  while  the 
other  moves  with  unit  velocity,  the  space  between  the  two 
being  filled  with  the  liquid"  (Maxwell).  The  reciprocal  of  the 
coefficient  of  viscosity  is  called  the  coefficient  of  fluidity. 

The  viscosity  of  water  decreases  as  the  temperature  rises 
and  at  any  temperature,,  t,  the  viscosity  is  represented  by  the 
empirical  equation.1 

77=2.989  (/  +  38.5)-'.4o  (76) 

127.  The  flow  of  liquid  through  a  capillary  tube  is  essen- 
tially of  the  nature  of  sliding  of  layer  over  layer.  The 
cylindrical  layer  in  immediate  contact  with  the  tube  remains 
fixed  or  at  least  has  no  motion  parallel  to  the  axis  of  the 
tube,  and  the  immediately  adjacent  layer  slides  over  it,  the 
next  layer  slides  over  the  second,  and  so  on  up  to  the  center 
of  the  tube.  (In  a  tube  of  greater  than  capillary  bore,  this 
is  not  so,  for  there  are  eddies  in  the  motion.  This'  distinction 
is,  in  fact,  the  best  definition  of  the  term  capillary.) 

Thus,  if  we  measure  the  force  causing  flow  through  the 
tube  and  the  rate  of  flow,  we  shall  be  in  a  position  to  deduce 
the  coefficient  of  viscosity  of  the  fluid.  If  m  is  the  mass 
of  liquid  of  density  p  that  flows  in  time,  T,  through  a  vertical 
tube  of  length,  /,  and  radius  of  bore,  r,  and  h  is  the  vertical 
distance  from  the  level  of  the  liquid  in  the  reservoir  above  the 
tube  to  the  lower  end  of  the  tube,  the  coefficient  of  viscosity  is 

(7    } 


~      Sml 

128.  Proof.  —  Suppose  all  the  liquid  in  a  capillary  tube  of  length  / 
and  radius  r  to  be  solidified  except  a  tubular  layer  of  mean  radius  x 
and  thickness  dx.  If  there  be  a  difference  of  pressure  p  (per  unit 
of  area)  between  the  two  ends,  the  solid  will  attain  a  steady  velocity 
such  that  the  viscous  resistance  just  equals  the  whole  difference 

1  Van  Laar,  Electrochemie,  p.  36. 


I2O  GASES,    VAPORS   AND    LIQUIDS. 

of  pressure  on  its  ends.      Hence  it  follows  from  the'  definition  of  the 
coefficient  of  viscosity  that 

zxxlvT) 

~~dx~   =™2p 
Hence  the  velocity  is 

_pxdx 

~    2  If) 

If  q  be  the  volume  of  the  core  that  flows  out  per  second 

pTtxsdx 


Suppose  now  another  layer  liquefied.  There  will  follow  a  further 
flow  represented  by  the  same  expression  but  with  a  different  value 
of  x.  Let  the  process  be  continued  until  the  whole  is  liquid,  then  the 
whole  flow  per  second,  V,  will  be  the  sum  of  all  the  values  of  q  for 
values  of  x  between  o  and  r.  Hence,  integrating, 


If  the  tube  be  vertical  and  the  flow  be  due  to  gravity  only,  instead 
of  p  we  must  put  gph.  If  m  is  the  mass  of  density,  p,  that  flows  out 
in  time  t,  m=Vpdt.  Hence,  by  substitution,  we  have  equation  76'. 
In  the  above  it  was  tacitly  assumed  that  the  liquid  adheres  to  the 
tube  without  any  slip.  If  there  were  any  slip,  the  overflow  would  be 
increased  by  it  and  the  above  expression  would  not  hold.  Poiseuille 
and  others  verified  the  above  formula  in  all  cases,  thus  showing  that 
no  slip  occurs. 

TABLE  XVI. 

Coefficient  of  Viscosity  (20°).  J 
Water  ........................................  oioo 

Mercury  .......................................  01  59 

Acetic  acid  ..............  ......................  0122 

Methyl  alcohol  ................................  00591 

Ethyl  alcohol  .................................  0119 

Ethyl  ether  ...................................  00234 

Benzol  ............................  ............  00649 

PROBLEMS  IV. 

1.  How  many  calories  are  required  to  heat  one  kilo  of  water  to 
100°  and  boil  it  away  at  this  temperature? 

2.  Estimate  the  density  of  liquid  ethylene  at  (a)  —  100°  (b)    —200° 
(c)  -273°  (§§120,  121  and  Table  LVI). 

3.  Calculate  the  force  exerted  by  surface  tension  upon  the  stem 
of  an  hydrometer  in  water,  if  the  diameter  of  the  stem  is  4  mm.  and 
if  (a)  the  water  is  at  o°;  (b)  if  it  is  at  20°. 

4.  How  high  will  benzol  rise  in  a  capillary  .2  mm.  internal  diam- 
eter at  30°? 

5.  Calculate   the  molecular  weight  of  liquid   benzol  at  10°  (§123 
and  Table  XV). 

6.  A  vertical  capillary  tube  .5  mm.  internal  diameter  and  10  cm. 


MEASUREMENT    OF    VISCOSITY. 


121 


long  is  inserted  in  the  bottom  of  a  reservoir  of  mercury.  The  level 
of  the  mercury  is  5  cm.  above  the  top  of  the  tube.  How  much 
mercury  will  flow  out  in  one  minute? 

EXPERIMENT  XI. 
Measurement  of  Viscosity. 

The  capillary  tube  should  be  about  20  cm.  long  and  its  bore  must 
be  as  nearly  circular  as  can  be  obtained. 

The  mean  radius  of  the  bore  can  be  best  determined  by  weighing 
the  amount  of  mercury  that  fills  a  measured  length  of  the  tube. 
The  tube  should  be  first  cleaned  by  attaching  it  to  the  end  of  a 
rubber  tube  at  the  other  end  of  which  is  a  hollow  rubber  ball,  and 
thus  drawing  through  it  and  forcing  out  a  number  of  times  (i) 
chromic  acid,  (2)  distilled  water,  (3)  alcohol,  and  finally  drying  it 
by  sucking  air  through  it.  Then  draw  into  the  tube  a  column  of 
clean  mercury  about  4  cm.  long  and  measure  its  length  as  accurately 
as  possible  by  a  comparator  (see  §28)  in  at  least  five  different  posi- 
tions in  the  tube  and  take  the  mean. 

The  mass  of  mercury  should  next  be  ascertained  by  weighing  it 
with  great  care  on  a  sensitive  balance.  The  mercury  should  not  be 
dropped  directly  on  the  scale  pan,  but  into  a  watch-glass  or  paper 
box  placed  on  the  scale  pan.  From  these  measurements  and  the 

density  of  mercury  at  the  temperature  of  observa-         . . 

tion  (see  Table  LIII)  the  diameter  of  the  bore  is 
obtained.  It  may  be  noted  that  since  it  is  r*  that 
is  used  in  the  formula  for  viscosity  and  r2  that  is 
obtained  directly  from  the  mercury  measurements, 
the  value  of  r  need  not  be  deduced. 

The  length  of  the  tube  may  be  measured  by  the 
comparator.  The  tube  is  then  attached  vertically 
by  a  rubber  connection  to  a  funnel  (Fig.  52)  and 
the  mass  of  liquid  that  flows  through  the  tube  in  a 
given  time  found  by  weighing  a  beaker  (i)  empty 
and  (2)  containing  the  water  that  has  passed.  The 
time  is  obtained  by  observing  a  clock  ticking 
seconds  or  a  chronometer.  It  is  evident  that  the 
greater  the  whole  time,  the  less  the  percentage 
error  in  time,  due  to  errors  in  observing  the  time 
of  starting  and  stopping,  and  so,  too,  the  greater 
the  whole  mass,  the  less  the  percentage  error  in 
weight.  Hence  the  time  and  the  mass  should  be 
sufficiently  great  to  make  the  percentage  errors  in 
them  less  than  those  in  I  and  r*.  To  prevent 
evaporation  from  the  beaker,  it  should  be  covered 
by  a  sheet  of  paper  pierced  by  a  hole  through 
which  the  tube  passes.  While  the  liquid  is  flowing, 
the  temperature  of  the  water  in  the  funnel  should 
be  noted. 


_^ 

T\ 

f 

T' 

1 

I 

i 

. 

FIG.  52. 


The  value  of  h  is  the  mean  of  its  values  at  the  beginning  and  end 
of  the  flow.  These  values  are  best  obtained  by  a  cathetometer 
(§29).  A  vertical  scale  mounted  near  the  tube  and  a  horizontal 
reading  telescope  may,  however,  be  used. 

1  Winkelmann,  1908,  1,  II,  p.  1397,  ff. 


122 


GASES,    VAPORS    AND    LIQUIDS. 


QUESTIONS. 

1.  Could    the    radius    be    found    satisfactorily    by    measurements 
with  a  micrometer  microscope  ?     Explain. 

2.  What  mass  of  this  liquid  would  flow  through  a  tube   i   mm. 
in  diameter  and  i  metre  long  in  one  hour,  if  it  were  similarly  attached 
to  the  funnel,  and  the  original  liquid  level  in  the  funnel  was  main- 
tained? 

3.  Two  flat  plates,  one  decimeter  square,  are  separated  by  one 
mm.  of  this  liquid.      What   force  would    be  required  to  move  one 
with  a  velocity  of  30  cm.  per  second,  the  other  being  at  rest  ? 


CHAPTER  II. 
THERMODYNAMICS.1 

129.  The  First  Law  of  Thermodynamics. — The  first  law  of 
thermodynamics  states  that  "when  work  is  transformed 
into  heat  or  heat  into  work,  the  quantity  of  work  is  mechan- 
ically equivalent  to  the  quantity  of  heat,"2  or 

W=JQ  (77) 

where  J  is  a  constant  called  the  mechanical  equivalent  of 
heat.  The  value  of/  will  be  determined  in  Experiment  XLV. 

If  W  is  expressed  in  ergs  and  Q  in  calories  (15°),  the  most 
accurate  determinations  of/  give  the  value  4. 187X1  o7.3  If 
W  is  in  joules  and  Q  in  calories,  /  is  4.187  (§7). 

When  a  quantity  of  heat  energy,  Q,  is  absorbed  by  a  body, 
the  equivalent  of  this  heat  energy  may  be  distributed  among 
three  forms  of  energy:  (a)  an  increase  in  the  kinetic  energy 
of  the  molecules  (that  is,  rise  of  temperature,  §"104)  =H; 
(b)  an  increase  in  the  potential  energy  of  the  molecules  (latent 
heat,  etc.)  =L;  and  (c)  external  work  (expansion  against 
external  pressure)  =  W. 

By  the  first  law  of  thermodynamics 

JQ  =  H+L+W  =  W-I  (78) 

where  —  /  represents  the  increase  in  the  internal  energy 
of  the  body,  as  W  represents  the  increase  in  the  energy  of 
the  surrounding  medium.  (The  signs  in  later  equations  are 
more  convenient  if  we  let  /  represent  the  decrease  of  internal 
energy  rather  than  the  increase.) 

1  General    references    for    Chapter    II:      Nernst,    "Thermodynamics    and 
Chemistry";  Maxwell,  "Theory  of  Heat";  Planck,  "Thermodynamics";  Winkel- 
mann,  1906,  vol.  iii;    Lewis,  " Thermodynamic  Chemistry,"  Proc.  Am.  Acad., 
T9°7>  43>  P-  259;    Haber,  "  Thermodynamics  of  Gas  Reactions." 

2  Maxwell,  Heat,  p.  152. 

3  Ames,  Rapports  Paris  Congres,  1900,  i,  p.  178. 

I23 


124  THERMODYNAMICS. 

130.  Specific  Heat. — The  specific  heat  of  a  body,  s,  is  the 
value  of  Q  for  one  gram  and  for  one  degree  rise  of  temperature. 
Therefore,  if  the  mass  of  a  body  is  m,  the  amount  of  heat 
required  to  change  its  temperature  from  /x  to  t2  is 

Q  =  ws  (/,-/,)  (79) 

The  specific  heats  of  various  solids  and  liquids  are  given 
in  Tables  XLIII,  XLIV,  and  XLV. 

The  molecular  heat  of  a  body,  5,  is  the  heat  required  to  give 
a  mass  numerically  equal  to  the  molecular  weight,  M,  a  rise 
of  temperature  of  one  degree,  or  S  =Ms. 

Similarly,  if  A  is  the  atomic  weight  of  an  element,  its 
atomic  heat  is  As. 

In  1819  Dulong  and  Petit1  announced  the  law  that  all  the 
solid  elements  have  the  same  atomic  heat.  More  careful  later 
measurements  have  shown  that  the  atomic  heats  are  not 
identical,  but  that,  with  a  few  exceptions,  the  values  at  or- 
dinary temperatures  lie  between  5.8  and  6.9.  The  mean 
value  is  about  6.4.  The  principal  exceptions  are  boron, 
carbon,  and  silicon,  the  atomic  heats  of  which  are  low.  At 
high  temperatures  these  three  elements  have  atomic  heats 
within  the  above  limits. 

131.  The  Specific  Heats  of  Gases. — The  thermal  expansion 
of  solids  and  liquids  is  so  slight  that  W  in  Equation  78  is  usually 
negligible.      With    gases,    however,    L   is   usually    negligible 
since  it  is  the  work  done  in  expanding  the  gas  against  the 
attraction  between  the  molecules  represented  by  the  term  a/v2 
of  Van  der  Waals'  equation  (Eq.  63),  but  the  coefficient  of  ex- 
pansion is  large,  and  W  may  therefore  be  large.     Since  the 
external  work  W  will  depend  on  the  conditions  under  which 
the  gas  changes  its  temperature,  the  specific  heat  is  indefinite 
unless  these  conditions  are  specified. 

If  the  volume  of  the  gas  is  kept  constant,  the  external  work, 
W,  is  zero.  Denoting  this  specific  heat  by  sv 

JQ=Jmsv(ta-tJ  =  -I  (80) 

1  Ann.  Chem.  Phys.,  1819,  x,  p.  395. 


SPECIFIC    HEATS.  125 

If  the  same  amount  of  gas  is  heated  through  the  same  range 
of  temperature,  and  is  allowed  to  expand  under  constant 
pressure,  p,  from  a  volume  Vj.  to  a  volume  v2 

JQ  =  Jmsp(ta-tI)=W-I  (81) 

Where  sp  is  the  specific  heat  under  constant  pressure. 

132.  Difference  of  Specific  Heats.  —  By  Equations  62  and  57 


Subtracting  Equation  80  from  Equation  81 

W  =  Jm(t2-tl)(sp-sv}. 
Since  (0a-0.)  =(*a-/i) 

SP-Sv=^  =  -3*Xl°'=i.9S$    (§107)    (82) 
J       4>I57  X  IO 

Or  the  difference  between  the  molecular  heat  of  any  gas  at 
constant  pressure  and  that  at  constant  volume  is  the  molec- 
ular gas  constant  in  calories  or  approximately  2  . 

133.  Ratio   of   Specific   Heats.  —  Dividing  Equation  81    by 
Equation  80  we  have 


s         _I+W     H+W 

—v=   ^rr    TT    r  (83) 

for  L  (Equation  78)  is  so  small  for  gases,  under  ordinary 
conditions,  that  we  will  neglect  it.  H,  which  represents 
the  increase  in  the  kinetic  energy  of  the  molecules,  may  be 
resolved  into  two  parts:  HIt  which  is  equal.  to  the  increase  in 
the  kinetic  energy  of  translation  of  the  molecules  =  1/2  m 
^_^)  (§104),  and  H2)  which  represents  the  gain  in  other 
forms  of  kinetic  energy,  for  example,  rotation,  vibration  of  the 
constituent  atoms,  etc.  Since 

(Eq.  41),  pv  = 


126 


THERMODYNAMICS. 


If  the  gas  is  monatomic  and  if  there  is  no  increase  in  the 
speed  of  rotation  of  the  molecules,  H2=o  and  ^  =  1.67.  The 
experimental  values  of  f  for  the  monatomic  gases,  mercury 
vapor,  argon,  and  helium  agree  well  with  this  number  (see 
Table  XVII).  The  more  complex  the  gas,  the  greater  is  the 
opportunity  for  other  forms  of  kinetic  energy  than  the  trans- 
lational,  and,  therefore,  as  the  number  of  atoms  increases,  we 
should  expect  that  H 2  would  increase  and  y  would  therefore 
decrease.  Table  XVII  shows  clearly  that  such  is  the  case. 

TABLE  XVII. 
Specific  Heats  of  Gases.1 


Temp. 

Sp 

Sp 

r=^ 

Argon  
Helium  
Mercury 

20° 
20° 

27  rO_~   £-6° 

.1205 

!-25 
0246 

.66 
.64 
66 

Hydrogen  
Nitrogen 

'/  3..    «3  3    _ 

0°-200° 
—  3O°-2OO° 

3.406 

24.4. 

•396 

4.0  s 

Oxygen  
Air  

0°-200° 
0°-200° 

.217 

.2^7  5 

.40 

.405 

Chlorine  
Iodine  
Bromine  
Water  . 

i9°-343° 
2oo°-377° 
85°-228° 
iio°-2  <o° 

•US 
.0336 

-0555 
.480 

•32 
.29 

•29 

.287 

Hydrogen  sulphide  .  .  . 
Carbon  dioxide  
A.mmonia 

I0°-200° 
100° 
2O°-2IO° 

•245 
.2I7 
<12 

.28 
,rf 

217 

Chloroform 

28°-n8° 

1  4-4- 

I  ^4 

Ethyl  alcohol  
Ether  

IIO°-220° 
70°-22<;0 

•453 
.480 

.14 
O7 

Benzol  .  .  

n6°-2i8° 

.•}  7  er 

.187 

134.  Specific  Heat  at  Different  Temperatures. — Le  Chatelier 
observed  that  the  specific  heats  of  all  gases  and  vapors  increased 
with  rise  of  temperature  but  at  different  rates.  He  also 
observed  that  if  the  molecular  specific  heat  at  constant 
pressure  (Msp)  was  exterpolated  for  the  absolute  zero, 
the  value  6.5  was  found  for  all  gases  and  vapors.  Therefore, 
by  §132,  the  molecular  heat  of  all  gases  and  vapors  at  constant 
volume,  at  the  absolute  zero,  is  4.5. 
*  Jiiptner,  I,  pp.  71-73. 


ADIABATIC    CHANGES.  127 

Table  XVIII  illustrates  the  variation  with  the  temper- 
ature. 

TABLE  XVIII. 
True  Molecular  Specific  Heats  at  Temperature  t.1 

(The  figures  below  are  for  constant  pressure;  for  constant  volume, 
subtract  2  (§132). 

Permanent  gases,      M sp  =  6.$  +  .0012  (£  +  273) 

(H2,02,N2,CO) 

Water  vapor,  M  5^  =  6.5  +  . 0058  (£  +  273) 

Ammonia,  M  sp  =  6. 5  +  . 0071  (£  +  273) 

Carbon  dioxide,          M  sp  =  6. 5  +  . 0074  (£  +  273) 
Nitrous  oxide,  M  sp  =  6.$  +  . 0089  (£  +  273) 

Benzol,-  M  s/>  =  6.5  +  .o5i     (£  +  273) 

The  mean  molecular  specific  heat  between  —273°  and  273°+*°  is 
equal  to  the  above  figures  if  the  coefficient  is  halved.  For  example, 
the  mean  molecular  specific  heat  of  water  vapor  at  constant  pressure 
between  — 273°  and  273°+^°  is 

6.5  +  . 0029  (2  +  273) 

135.  Adiabatic  Changes. — If  we  combine  Equations  78,  80, 
and  62  we  have 

JQ  =  Jmsv  (t2  -tj+p  (v2  -  v,)  (85) 

In  §97  we  have  considered  the  relation  between  the  pressure 
and  volume  when  the  temperature  is  constant.  We  shall 
now  consider  the  relations  between  all  three  quantities  when 
no  heat,  Q,  either  enters  or  leaves  the  body.  Such  changes 
are  called  adiabatic, 

The  simplest  method  is  to  consider  first  very  small  changes  and 
later  find  the  relation  for  finite  changes  by  integration. 

.'.  O=Jmsvdt  +  pdv  (86) 

We  shall  first  find  the  relation  between  p  and  v  and  therefore  we 
shall  eliminate  dt.  Differentiating  Equation  57,  we  have 


R-jfdB  (86') 

.•  ^^  (pdv+vdp)+pdv  =  o 
K. 


dv  +  JMsvvdp  = 
1  LeChatelier,  Zeit.  phys.  Chem.,  1887,  i,  p.  456. 


128  THERMODYNAMICS. 

But 

JMsv  +  R=JMsp  (Equation  82) 
Hence,  dividing  by  JM 

sppdv  +  svvdp  =  o 
vdp       _  _sp     _ 

136.  Velocity  of  Sound. — This  equation  is  the  basis  of  a 
method  of  determining  y.  The  velocity  of  elastic  waves  in 
any  medium  is 

I  <«> 

where  p  is  the  density  and  E  is  the  coefficient  of  elasticity  or 
the  ratio  of  stress  to  strain.  If  the  stress  dp  is  applied  to  a 
gas,  the  strain,  or  change  in  volume  per  unit  volume,  is 


Equation  87  shows  that  E  is  equal  to  yp  if  no  heat  leaves  or 
enters  the  gas.  Such  is  the  case  with  the  rapid  vibrations  of 
sound  waves.  Their  velocity  is  therefore  given  by  the  formula 

(880 

Since  u,  p,  and  p  may  easily  be  measured  with  high  accuracy, 
this  indirect  method  gives  accurate  values  of  y.  Since  p/p 
is  proportional  to  6  (Equation  53),  the  velocity  at  any  tem- 
perature t  is  related  to  u0,  the  velocity  at  o°  by  the  equation 


r    ;      '   (88"} 

EXPERIMENT  XII. 
Velocity  of  Sound  by  Kundt's  Method.     Ratio  of  Specific  Heats. 

A  glass  tube,  A  G,  about  a  metre  long  and  about  3  cm.  internal 
diameter  is  closed  at  one  end  by  a  tight-fitting  piston,  C,  and  at  the 
other  end  by  a  cork  through  which  passes  a  glass  tube  having  at 
one  end  a  loosely-fitting  cardboard  disk,  D  (Fig.  53).  The  glass 
tube  should  be  about  a  metre  long.  A  little  dry  lycopodium  powder 
is  sprinkled  in  the  tube,  the  stopper  at  G  is  loosened,  and  a  current 


EQUATIONS    OF   ADIABATIC.  1 29 

of  air,  dried  by  passage  through  several  drying  tubes,  is  slowly  forced 
through  the  hollow  rod  of  the  piston,  C.  The  stopper  at  G  is  then 
replaced  and  the  glass  tube,  F,  is  held  at  the  center  and  stroked 
longitudinally  with  a  damp  cloth.  The  piston,  C,  is  adjusted  until 
the  powder  collects  in  the  sharpest  attainable  ridges.  These  ridges 
will  appear  where  the  pressure  changes  are  least;  that  is,  at  the 
loops.  Measure  carefully  the  distance  between  two  extreme  ridges 
and  divide  by  the  number  of  segments  into  which  the  tube  is  divided. 
This  distance  (between  two  loops)  is  a  half-wave  length,  of  the 


E        '     1            1 

|  C..^                  F 

1           1 

|  c  |  _n 

A        C 

D                      G 

FIG.  53. 

waves  in  the  tube.  Disturb  the  powder  and  make  a  new  adjust- 
ment of  the  piston,  C,  and  a  new  measurement  of  the  half -wave  length. 
Make  a  third  repetition  of  the  adjustments  and  readings. 

Fill  the  tube  with  another  dried  gas,  for  example,  carbon  dioxide, 
illuminating  gas,  hydrogen,  oxygen,  or  hydrogen  sulphide,  and 
determine  the  half -wave  length.  If  n  is  the  constant  pitch  of  the 
note  emitted  by  the  glass  tube  and  >i  is  the  wave  length  in  the  gas 

(88'") 

-M-2  2 

Since  the  velocity  changes  at  the  same  rate  with  change  of  tem- 
perature in  all  gases  (Eq.  88"),  the  velocity  of  sound  or  compressional 
waves  at  zero  degrees  in  any  other  gas  than  air  can  be  calculated 
from  the  ratio  of  the  wave  lengths  at  a  common  temperature,  and 
the  velocity  in  air  at  zero  degrees  (33,200  cm.  per  second). 

From  the  velocity  of  sound  at  zero  degrees  in  the  gases  other  than 
air,  calculate  the  ratio  of  specific  heats,  f.  Table  LI  I  gives  the 
densities  of  the  more  common  gases  and  vapors  at  zero  degrees  and 
a  pressure  of  76  cm.  of  mercury  =  1013200  dynes  per  square  centi- 
meter. 


QUESTIONS. 

1.  Calculate    (a)    the   velocity   of   compressional   waves   in   glass, 
(b)  the  elasticity  E.  (Notice  that  each  end  of  the  glass  rod  must  be  a 
loop,  and  the  center  a  node.     The  density  of  glass  can  be  obtained 
from  Table  LV). 

2.  Why  must  the  glass  rod  be  set  in  longitudinal  vibration? 

3.  Why  does  the  powder  collect  at  the  loops? 

137.  Equations  of  Adiabatic. — We  shall  show  that  the  equa- 
tion for  an  adiabatic  change  is 

$=£)r  (89) 


I3O  THERMODYNAMICS. 

Let  us  integrate  Equation  87,  between  the  limits  pi  and  p2,  vlt  and  v2 


or 


.  _£i    /Mr 

'Pi        \VaJ 


If  we  draw  adiabatic  curves  in  a  pressure  volume  diagram, 
they  will  be  steeper  than  the  isothermal  curves  for  which  (38) 

h.=^L 

for  Y  is  always  greater  than  unity. 

We  shall  next  show  that  the  relation  between  the  pressure 
and  the  absolute  temperature  in  an  adiabatic  change  is 

8 


^     /M 

0,       \PJ 


r  (90) 

Substitute  for  pdv  in  Equation  86  its  value  from  Equation  86'. 


.'.  mMspdO  =  MR-j-j- dp  (Equations  82  and  57) 

dd_  _    R     dp_  _  sp-Sy  dp_  _  y-i  dp_ 
0    ~  Msp    p  ~      sp        P  ~     T       P 
Integrating  between  the  limits  Olt  and  02,  pi,  and  p? 

0  2  f I  p2 

la^  "~rln?r 


02  I    p2\ 

*:TZ-(K) 


Also,  by  Equation  89 


0,      \v 


LIQUEFACTION    OF    GASES.  131 

Liquefaction  of  Gases. 

138.  Method  of  Claude. — We  saw  in  §115  that,  in  order  to 
liquefy  a  gas,  the  temperature  must  be  reduced  below  the 
critical  temperature  and  a  suitable  pressure  must  be  applied. 
The  most  successful  apparatus  for  liquefying  gases  is  that  of 
Claude1  where  the  gas  is  compressed  and  then  cooled  by 
adiabatic  expansion.  The  essential  features  of  the  apparatus 


Liq.  Air.  40  at.,-  140 


FIG.  54 


are  represented  in  Fig.  54.  The  air,  compressed  to  40  atmos- 
pheres pressure  by  a  compressor  (not  shown) ,  passes  through 
the  tube  (actually  a  worm)  A,  and  divides  at  B.  A  portion 
enters  the  cylinder  through  a  suitable  valve,  and,  expanding, 
forces  out  the  piston,  thereby  doing  work  and  cooling  itself 
(Equation  78).  This  very  cold  air  circulates  through  the 
liquefier  L  where  it  liquefies  the  portion  of  the  compressed  air 
which  entered  at  B,  and  then,  emerging  along  the  outside 

1  Comptes  Renclus,  1900,  II,  p.  500;  1902,  I,  p.  1568;  1905,  II,  p.  762;  p.  823; 
Journal  de  Physique,  1906,  p.  5. 


132  THERMODYNAMICS. 

of  A  ("regenerator"),  cools  the  entering  air  and  returns  to  the 
compressor.  Its  pressure  after  leaving  D  is  not  far  from 
atmospheric  and  its  temperature  is  below  — 140°,  but  not  as  low 
as  —  190°  at  which  temperature  air  liquefies  under  atmospheric 
pressure.  The  highly  compressed  air  in  the  inner  tubes  of  the 
liquefier  is,  of  course,  liquefied  at  a  higher  temperature  (about 
—  140°).  The  motor  D  restores  about  one-fourth  of  the 
power  consumed  by  the  compressors.  •  Nearly  one  litre  of 
liquid  air  is  obtained  per  horse-power  hour. 

139.  Method  of  Linde. — Previous  to  the  development  of  the 
adiabatic  method  of  liquefying  gases,  the  most  efficient  method 
was  that  of  Linde.     In  this  method  the  gas  was  allowed  to 
expand  through  a  small  opening  from  a  pressure  of  about 
200  atmospheres  to  about  20.     The  work  done   against   the 
attraction    between    the    molecules    (a/v2    term  in   Van   der 
Waals'  equation)  cools  the  gas  until,  by  a  regenerative  process 
similar  to  that  of  A  and  M  of  Fig.  54,  the  critical  temperature 
is  reached  and  the  gas  is  liquefied. 

140.  Separation  of  Oxygen. — The  technical  importance  of 
liquid  air  arises  from  the  possibility  of  separating  the  oxygen 
and  nitrogen  by  fractional  distillation. 

Under  atmospheric  pressure,  nitrogen  boils  at  — 194°  and 
oxygen  at  — 180.5°.  Therefore,  the  vapor  above  liquid  air  is 
excessively  rich  in  nitrogen  while  the  remaining  liquid  is 
very  rich  in  oxygen.  The  two  gases  may  be  almost  completely 
separated  by  the  apparatus  sketched  in  Fig.  55.  (Apparatus 
of  Linde  and  Claude.) 

The  compressed  and  cooled  air  enters  at  A  and  as  it  rises 
to  B  much  of  the  oxygen  and  some  of  the  nitrogen  is  condensed 
by  the  liquid  oxygen  outside  the  tubes,  and  falls  to  C.  The 
remainder,  which  is  almost  pure  nitrogen,  is  further  condensed 
as  it  comes  down  through  the  tubes  surrounded  by  liquid 
oxygen,  and  collects  in  D.  E  is  a  tower  about  three  metres 
high,  filled  with  glass  marbles  or  similar  bodies.  The  pressure 
in  C  forces  the  impure  liquid  oxygen  up  through  the  tube,  F, 
which  discharges  into  the  middle  of  the  column.  As  the  liquid 
trickles  down,  the  nitrogen  gradually  evaporates  and  rises, 


LIQUEFACTION    OF    HELIUM. 


133 


and  the  purified  oxygen  finally  collects  in  H,  whence  it  is 
drawn  off  by  the  tube,  K.  The  liquefied  impure  nitrogen 
discharges  at  the  top  of  the  column.  Since  the  vapors  arising 
from  the  oxygen  in  H  and  E  are 
at  a  higher  temperature,  practi- 
cally all  the  nitrogen  will  evap- 
orate and  escape  at  N. 

141.  Liquefaction  of  Helium.— 
On  July  10,  1908,  Kamerlingh- 
Onnes    liquefied    helium   which 
had   hitherto   resisted   liquefac- 
tion.      He    found    the    critical 
temperature    about    5     degrees 
absolute,  the  boiling  point  about 
4.3°   absolute   and   the    critical 
pressure   between  2    and    3    at- 
mospheres.    The  density  at  the 
boiling  point  was  .15.     By  boil- 
ing liquefied   helium   under   re- 
duced pressure,  he  attained  the 
extremely  low  temperature  of  3 
degrees  absolute.1 

142.  Clement   and   Desormes' 
Method    of    Measuring    7-    (Gay 
Lussac's     Modification) . 2 — The 
gas  is  compressed  into  a  vessel 
until  the   pressure   has  a  value 
which  we  will  designate  by  pt. 
The  vessel  is  then  opened  for  an 
instant,  and  the  gas  rushes  out 
until  the  pressure  inside  falls  to 
the    atmospheric    pressure,    p0. 

This  expansion  may  be  made  so  sudden  that  it  is  practi- 
cally adiabatic  and  the  temperature  of  the  gas  will  therefore 
fall.  After  the  vessel  has  been  closed  for  a  few  minutes,  the 


98%  0, 


A»r 


1  Comm.  Phys.  Lab.,  Leyden,  No.  108. 

2  Jour,  de  Phys.,  1819,  Ixxxix,  p.  333. 


134 


THERMODYNAMICS. 


gas  will  have  warmed  to  the  room  temperature  and  the  pres- 
sure, p2,  will  be  above  that  of  the  atmosphere.  Consider  one 
gram  of  the  gas.  During  the  adiabatic  expansion,  its  volume 
changed  from  vl  to  v2,  according  to  Equation  89  or 


Since  the  initial  and  final  temperatures  are  the  same,  and 
since  the  volume  remains  v2  while  the  gas  is  warming,  and 
the  pressure  is  rising  from  p0  to  p2, 


P 


•'•r  = 


(92) 


The  successive  operations  will  be  evident  from  a  study  of 
Fig.   56- 


FIG.  56. 

Lummer  and  Pringsheim x  improved  the  method  by  observ- 
ing, with  a  bolometer  strip  .0006  mm.  thick,  the  fall  in  tem- 
1  Wied.  Ann.,  1898,  Ixiv,  p.  555. 


RATIO    OF    SPECIFIC    HEATS.  135 

perature  during  the  adiabatic  expansion,   and,   knowing  pL 
and  p0,  they  calculated  f  by  Equation  90. 

PROBLEMS  V. 

1.  A  copper  stirrer  weighing  30  gr.,  turning  in  200  c.c.  of  dilute 
solution    requires    20    watts   (§5).     Assuming    that    practically    all 
the  power  is  absorbed  by  the  viscous  resistance  between  the  solution 
and  the  stirrer,  calculate  (a)  the  total  amount  of  heat  produced  in 
one  minute;  (b)  the  rise  of  temperature,  if  there  are  no  heat  losses. 

2.  The  specific  heat  of  indium  is  .0569.      What  is  the  approximate 
atomic  weight  ? 

3.  How  much   heat  is  required  to  warm  a  body  of  air  which 
occupies  one  cubic  meter  at  o°  and  76  cm.  from  o°  to  100°,  (a)  the 
volume    being    constant?      (b)    the    pressure    being    constant?      (c) 
Find  the  corresponding  amounts  of  heat  for  hydrogen. 

4.  Calculate   the    mechanical   equivalent    of   heat    from   the   two 
specific    heats    of    oxygen  and  the  gas  constant    (Equation   82). 

5.  How  much  heat  is  required  to  warm,  from  o°  to  1000°,  5  grams 
of    (a)    oxygen    at    constant    volume?      (b)    hydrogen    at    constant 
pressure?      (Table  XVIII.) 

6.  How  much  heat  is  required  to  warm,  from  950°  to  1000°,  100 
grams  of  (a)  oxygen  at  constant  volume ;  (b)  hydrogen  at  constant 
pressure?      (Table  XVIII.) 

7.  The  velocity  of  sound  in  ammonia  gas  at  o°  is  415  meters  per 
second.     Calculate  the  ratio  of  specific  heats  (find  the  density  by 
Equation -56). 

8.  A  mass  of  nitrogen  which  occupies  one  litre  at    100°   and  a 
pressure  of  200  cm.  of  mercury,  expands  adiabatically  to  one  and  a 
quarter  litres.     Calculate  (a)  final  pressure;  (b)  final  temperature. 

EXPERIMENT  XIII. 
Measurement  of  the  Ratio  of  Specific  Heats  by  Adiabatic  Expansion. 

A  large  carboy  is  mounted  in  a  wooden  case  and  may  be  sur- 
rounded with  cotton  batting.  The  neck  is  closed  with  a  rubber 
stopper  through  which  passes  a  T-tube  connected  on  one  side  with 
a  compression  pump  (e.g.  a  bicycle  pump),  and  on  the  other 
side  with  a  manometer  containing  castor  oil.1  A  large  glass  tube, 
which  may  be  closed  by  a  rubber  stopper,  also  passes  through  this 
large  stopper.  A  little  sulphuric  acid  in  the  bottom  of  the  carboy 
keeps  the  air  dry.  A  very  fine  copper  wire  and  a  very  fine  constantin 
wire  pass  tightly  through  minute  holes  in  the  stopper  and  meet  at 
the  center  of  the  carboy,  in  a  minute  drop  of  solder. 

The  air  in  the  carboy  is  compressed  until  the  difference  in  pressure 
is  about  40  cm.  of  oil  ( —  fi  —  p0).  The  tube  connecting  with  the  pump 
is  closed,  and,  after  waiting  about  1 5  minutes  to  allow  the  air  inside 
to  regain  its  initial  temperature  (as  shown  by  the  pressure  becoming 
constant),  the  ends  of  the  oil  column  are  carefully  read.  The  carboy 
is  now  carefully  surrounded  with  cotton  batting,  which  may  have 
been  removed  to  facilitate  cooling.  The  air  inside  is  momentarily 

1  The  density  of  castor  oil  is  about  .97,  but  it  should  properly  be  determined 
(§36). 


i36 


THERMODYNAMICS . 


allowed  to  return  to  atmospheric  pressure  by  removing,  for  about  one 
second,  the  rubber  stopper  from  the  glass  tube.  After  waiting  until 
the  air  inside  has  assumed  the  room  temperature  (shown  by  the 
pressure  becoming  constant),  the  final  pressure  p2  is  determined. 
The  cotton  wool  had  better  be  removed  during  this  stage.  Connect 
the  wires  to  a  calibrated  galvanometer  (§49),  apply  the  initial 
compression  plt  and  observe  the  reading  of  the  galvanometer  when 
it  has  become  steady.  Remove  the  stopper  as  before  (for  not  over 
one  second),  replace  the  stopper,  and  observe  the  galvanometer 
reading.  The  proper  reading  to  record  is  the  fairly  steady  deflection 
which  is  attained  immediately  after  the  stopper  is  removed.  There 
are  liable  to  be  rapid  fluctuations  which  should  be  disregarded,  and 


FIG.  57. 


of  course  the  temperature  does  not  long  remain  steady,  owing  to 
heating  or  cooling  from  the  outside.  Record  as  before  the  final 
pressure  p2.  Repeat  several  times,  starting  with  the  same  initial 
pressure  pI.  Record  the  temperature  of  the  room,  t,  and  p0,  the 
height  of  the  barometer  (§§40,  41). 

Calculate  y,  the  ratio  of  specific  heat  by  Equation  92. 

Calculate  the  change  of  temperature  from  the  galvanometer 
deflections  and  the  constants  ( §49).  Compare  the  result  with  #i — 60, 
where  0,  =273  +  t  and  00  is  calculated  by  Equation  90.  Unless 
exceedingly  fine  wire  is  employed,  the  heat  capacity  of  the  wire  is 
relatively  so  great  that  the  thermocouple  will  not  show  the  full 
change  of  temperature. 


THE    SECOND    LAW    OF    THERMODYNAMICS.  137 

Draw  a  curve  with  volumes  as  abscissae,  and  pressures  as  ordinates, 
which  will  represent  the  changes  in  this  experiment. 

(Let  specific  volumes,  i.e.,  volumes  of  one  gram,  be  abscissae. 
Calculate  from  Table  LII  and  Equation  53  the  specific  volumes  cor- 
responding to  the  room  temperature  and  p0,  pi,  and  p2,  and  draw  the 
corresponding  isothermal.  Draw  the  horizontal  line  corresponding 
to  p0.  Draw  a  vertical  through  the  point  corresponding  to  p2  on  the 
above  isothermal.  The  intersection  of  these  two  straight  lines  will 
evidently  be  p0,  v2. 

QUESTIONS. 

1.  Do  you  see  any  objection  to  an  initial  exhaustion  of  the  gas 
in  place  of  the  compression? 

2.  What  are  the  advantages  and  disadvantages  of  a  large  opening? 
Short  time  of  opening?     Castor  oil  manometer? 

3.  How  would  an  aneroid  manometer  be  preferable  in  this  experi- 
ment to  a  liquid  manometer? 

The  Second  Law  of  Thermodynamics. 

143.  The  second  law  of  thermodynamics  is  a  statement  of 
the  conditions  which  govern  the  transformation  of  heat  energy 
into  mechanical  energy.     The  first  law  states  that,  if  such  a 
transformation  takes  place,  the  two  are  equivalent,  but  does 
not  state  whether  or  not  such  a  transformation  is  possible. 
The  following  is  Clausius'1  statement  of  the  second  law. 

"It  is  impossible  for  a  self-acting  engine,  unaided  by  any 
external  energy,  to  convey  heat  from  one  body  to  another  at 
a  higher  temperature." 

Kelvin's  statement  of  the  law  is,  "It  is  impossible,  by 
means  of  inanimate  material  energy,  to  derive  mechanical 
effect  from  any  portion  of  matter  by  cooling  it  below  the  tem- 
perature of  the  surrounding  objects."1 

This  law  will  be  best  understood  by  considering  a  concrete 
example  of  the  transformation  of  heat  energy  into  mechanical 
energy  or  vice  versa.  In  order  to  measure  the  net  energy 
changes,  the  working  substance  should  be  brought  back  to  its 
original  condition,  or,  as  it  is  termed,  the  working  substance 
must  be  carried  through  a  cycle. 

144.  Carnot's    Cycle. — The    simplest    case    to    consider    is 
Car  not 's  cycle.1 

1  "The  Second  Law  of  Thermodynamics,"  Magie. 


138 


THERMODYNAMICS. 


A  given  mass  of  gas  expands  in  succession  isothermally 
and  adiabatically  and  is  then  isothermally  and  adiabatically 
compressed  to  its  original  state. 

Suppose  we  have  in  a  cylinder  with  a  movable  piston  a 
mass,  m,  of  a  permanent  gas,  such  as  oxygen,  at  a  pressure  pl, 


FIG.  58. 

and  a  volume  vlt  represented  by  the  point  A  in  the  diagram, 
Fig.  58.  Allow  the  gas  to  expand  in  a  thermostat  to  a  volume 
v2  while  0!  calories  of  heat  are  supplied  to  maintain  the 
temperature  constant  at  its  initial  value  0^.  By  the  first 
law  of  thermodynamics,  the  work  done  by  the  gas  is 


v2 

V2  S* 

i=7Qi=  I  pdv  =  R'mOl    1 

J 


(93) 


Now  surround  the  cylinder  by  a  non-conducting  envelope  and 
allow  it  to  expand  adiabatically  to  the  volume  v3t  at  the 
temperature  (absolute)  Oa.  The  work  done,  represented  by 

1  Sur  la  puissance  motrice  du  feu.     Paris,  1844.     Reprinted  in  "The  Second 
Law  of  Thermodynamics,"  Magie. 


139 

the  area  below  B  C,  is  equal  to  the  loss  of  energy  of  the  gas, 
since  there  is  no  transfer  of  heat. 

The  non-conducting  envelope  is  now  removed  and  the  cylin- 
der is  placed  in  a  thermostat  at  this  temperature  02.  The 
gas  is  compressed  until  the  volume  is  v4.  In  order  to  keep 
the  temperature  constant,  the  thermostat  must  have  absorbed 
an  amount  of  heat  Q2,  which  is  equivalent  to  the  work  done 
in  compressing  the  gas,  or 


J^ 
pdv  =  R'mln^ 


(94) 


VT, 

(The  minus  signs  signify  that  the  heat  is  emitted  by,  and  the 
work  is  done  upon  the  working  substance.) 

The  volume  v4  at  which  the  compression  was  stopped  was 
so  chosen  that  if  at  this  point  (D)  the  cylinder  is  removed 
from  the  thermostat  and  is  surrounded  by  a  non-conducting 
envelope,  a  further  adiabatic  compression  restores  the  gas  to 
its  original  state,  where  the  pressure,  volume,  and  temperature 
were,  respectively,  ply  vlt  and  0I. 

Dividing  Equation  93  by  Equation  94,  and  remembering 
that  a  negative  logarithm  is  the  logarithm  of  the  reciprocal, 


But  by  Equation  91 


;•<«-.*  (95) 

which  may  also  be  written,  by  the  laws  of  proportion, 

t~T  =  ° 
Q*=&-°^±-.w.  •   (96) 


140  THERMODYNAMICS. 

For  the  net  work  done  in  the  cycle,  W  =Wl  —  W2,  is  equal 
to  J  times  the  net  heat  absorbed  (Qt  —  Q2) 


•••  W^JQ-'-  (97) 

145.  Isothermal  Cycle.  —  If  6l=62,  W=o,  or  the  total  work 
performed  in  an  isothermal  cycle  is  zero. 

146.  Reverse  Cycle.  —  If  we  performed  the  operation  in  the 
reverse  order  A  D  C  B  A  ,  we  would  evidently  obtain  the  same 
relations  between  Q2,  which  would  now  be  the  heat  absorbed  by 
the  gas  during  the  expansion  D  C  and  Qx,  which  would  be  the 
heat  emitted  during  the  compression  B  A.     In  this  case  W 
would  be  the  net  work  done  upon  the  working  substance. 

Notice  that,  in  accord  with  the  above  statements  of  the 
second  law,  work  is  required  to  transfer  heat  from  a  low  tem- 
perature to  a  high  temperature  (above  reverse  process)  and 
that  when  heat  is  absorbed  at  a  low  temperature  no  work  can 
be  obtained  from  this  heat,  but  on  the  contrary,  work  must 
be  expended,  if  the  working  substance  is  to  be  returned  to  the 
initial  state. 

147.  Reversible    Processes.  —  Before    discussing   Equations 
95-97,  we  shall  show  that  they  are  independent  of  the  nature 
of  the  working  substance  or  operations,  provided  all  the  steps 
are  reversible.     A  process  is  reversible  if  at  each  instant  the 
system  is  in  equilibrium,  so  that  a  very  small  change  in  the 
conditions   will    change   the    direction    of   the    process.     For 
example,  if  the  expansion  A  B  (Fig.  58)  is  reversible,  a  very 
small  increase  in  the  external  pressure  will  cause  the  gas  to 
compress  and  thereby  emit  heat  instead  of  absorbing  heat. 
Similarly,  a  slight  fall  in  the  temperature  of  the  thermostat 
would  change  an  absorption  of  heat  along  A  B  to  an  emission 
with   an    accompanying    reversal    from    expansion    to    con- 
traction. 

If  Equations  95  and  97  do  not  hold  for  all  working  substances 
and  reversible  processes,  we  will  choose  a  system  or  "engine" 
which  absorbs  the  same  amount  of  heat,  Qx,  at  temperature 
0I  as  the  engine  which  we  have  just  considered,  but  gives 


EFFICIENCY.  141 

up  a  less  amount  of  heat,  Q2',  at  temperature  02.  By  the 
first  law  of  thermodynamics,  it  must  do  a  greater  amount 
of  work  =/  (Qi  — QaO-  Suppose,  for  example,  that  Qa'  = 
.9  Q2.  Allow  this  engine  to  make  ten  cycles  between  Ol  and  02. 
ioQi  units  of  heat  will  be  absorbed  at  temperature  6T, 
ioQ2'  =  902  will  be  emitted  at  temperature  62,  and  an  amount 
of  work/  (loQj—  902)  will  be  gained.  Now  run  the  original 
gas  engine  through  ten  cycles  in  the  reverse  direction.  It  will 
absorb  ioQ2  units  of  heat  at  the  low  temperature,  emit  loQj 
units  at  the  high  temperature  and  will  require  for  the  operation 
10  J  (Qi~  82)  units  of  work.  The  net  result  will  therefore  be 
a  gain  ofJQ2  units  of  work,  the  equivalent  of  which  has  come 
from  the  cold  body.  This  would  be  contrary  to  the  second  law 
of  thermodynamics,  and  therefore  the  second  type  of  engine 
does  not  exist. 

By  a  similar  consideration  of  other  imaginary  engines  we 
can  show  that  Equations  95-97  apply  to  every  reversible 
process  by  which  heat  energy  is  transformed  into  other  forms 
of  energy. 

148.  Efficiency. — According  to  Equation  97,  the  work  which 
is  obtainable  from  a  quantity  of  heat  Q,  under  the  rngst  perfect 
conditions,  is  the  fraction 


9, 

of  the  energy  of  the  heat.  This  fraction  is  called  the  efficiency 
of  the  engine.  If  a  steam  engine  receives  steam  at  a  tempera- 
ture of  140°  and  exhausts  into  the  air  at  a  temperature  of  105°, 
the  theoretical  efficiency  is  .085.  If  high-pressure  steam  at 
250°  is  used  and  the  exhaust  is  into  a  condenser  at  tempera- 
ture 4d°,  the  efficiency  is  raised  to  .40. 

149.  Kelvin's  Absolute  Scale  of  Temperature. — Kelvin's 
absolute  scale  of  temperature  is  based  upon  the  converse  of 
Equation  97.  The  temperature  of  a  body  is  measured  by 
the  efficiency  of  a  reversible  engine  working  between  this 
temperature  and  a  standard  temperature.  The  temperature 
interval  between  ice- water  and  steam  at  76  cm.  pressure  is 


142  THERMODYNAMICS. 

taken  as  100.  Equation  97  and  the  preceding  derivation 
shows  that  this  scale  (which  is  called  absolute  because,  as  we 
have  seen,  the  efficiency  is  independent  of  the  working  sub- 
stance) coincides  with  the  absolute  gas  thermometer  scale  of 
a  gas  which  obeys  Boyle's  and  Gay  Lussac's  laws.  No  gas 
completely  fulfills  these  conditions,  but  the  deviations  can  be 
obtained  experimentally  and  thus  a  table  of  corrections  can  be 
made  by  which  any  gas  thermometer  can  be  reduced  to  Kelvin's 
scale.  The  temperature  of  ice  and  water  is  273.13°  absolute.1 

150.  Entropy. — When  a  body  receives  a  quantity  of  heat 
Q  at  the  absolute  temperature  0  it  is  said  to  gain  an  amount  of 
entropy  numerically  equal  to  the  quotient  of  the  two  numbers. 

If  we  designate  the  gain  in  entropy  by  T? 

9— f-  (98) 

The  actual  amount  of  entropy  in  a  body  is  as  difficult  to 
define  (and  relatively  as  unimportant)  as  the  actual  quantity 
of  energy.  We  are  only  interested  in  the  changes  in  these 
quantities. 

Equation  95  states  that  a  reversible  cycle  produces  no 
change  in  the  entropy  of  the  system  composed  of  the  working 
substance  and  the  thermostats. 

151.  Irreversible    Cycles. — The    reversible    cycles   thus    far 
considered  are  unrealizable  in  practice.     The  pressure  of  the 
working  substance  must  be  greater  than  the  external  pressure 
in  order  to  overcome  inevitable  friction,   and  therefore  the 
work  obtained  is  less  than  is  represented  by  the  area  A  B  C  D 
of  Fig.     58.     The    temperature  of   the    thermostat  must  be 
slightly  above  #r  in  order  that  the  heat  may  flow  into  the  gas, 
and  for  the  same  reason  the  temperature  of  the  thermostat 
which   receives  the   heat    Q2   must  be   somewhat  below   02. 
Moreover,  some  of  the  heat  transferred  will  invariably  escape 
to  other  bodies. 

1  Buckingham  has  published  an  excellent  summary  of  the  deviations,  and  of 
the  correction  at  different  temperatures,  in  Bull.  Bureau  of  Standards,  1907, 
P-  237- 


IRREVERSIBLE    CYCLES. 


. ? 


or 


(99) 


(990 


Expressed  in  words,  the  entropy  of  any  irreversible  system 
tends  to  increase. 

Since  all  actual  systems  are  irreversible,  we  have  the  general 
laws,  "The  entropy  of  the  universe  tends  to  increase,"  "The 
energy  of  the  universe  is  constant"  (Clausius). 

If  we  are  more  directly  interested  in  the  heat  energy 
than  in  the  mechanical  energy,  a  temperature-entropy 
diagram  is  often  more  convenient  than  a  pressure-volume 
diagram.  Fig.  59  gives  the  temperature-entropy  diagram 

9 


FIG.  59. 

of  the  Carnot  cycle  of  Fig.  58.  Since  the  initial  value  of 
T)  is  unimportant,  it  is  taken  as  zero.  The  ratio  of  the  numeri- 
cal values  of  the  areas,  A  B  C  D,  in  the  two  figures  should 
equal  the  mechanical  equivalent,  for 


The  actual  cycles  of  heat  engines  are  not  likely  to  be  made 
up  of  isothermal  and  adiabatic  steps  as  is  Carnot's  cycle,  but 


144 


THERMODYNAMICS. 


any  cycle  may  be  resolved  into  infinitesimal  isothermal 
and  adiabatic  cycles,  as  is  illustrated  in  Fig.  60  for  a  portion 
of  a  circular  cycle.  The  long  portions  are  common  to  two 
cycles  and  neutralize  each  other.  The  short  portions  may 
be  made  to  differ  infmitesimally  from  the  original  curve. 


FIG.  60. 


152.  Free   Energy   Equation. — If   the   cycle   itself  is   very 
small  we  may  write  Equation  97  as 


(100) 


or,  for  infinitesimal  changes,  we  may  use  the  notation  of  the 
differential  calculus 


or 


=  O--  =  W-I   (Eq.  78) 


(100') 


(101) 


153.  Total,    Free    and    Latent    Energy. — Helmholtz    called 

W,  which  is  the  reversible  and  therefore  the  maximum  work, 

the  decrease  in  free  energy,  and  he  gave  the  name  of  latent 

energy  to  JQ  =  W—I,  which  is  the  excess  of  the  mechanical 


APPLICATIONS    OF    THE    SECOND    LAW.  145 

energy  or  free  energy  produced  over  the  decrease  in  total 
internal  energy  I. 

We  will  consider  a  few  examples.  When  a  reaction  occurs 
at  constant  volume,  W,  the  work  or  change  in  free  energy,  is 
zero,  and  the  decrease  /  of  the  total  internal  energy  is  measured 
by  the  latent  energy,  —JQ,  which  is  the  heat  emitted. 

Similarly,  during  a  change  of  state,  for  example,  the  freezing 
of  water,  W  may  be  negligible  compared  with  the  decrease 
of  internal  energy  /  or  its  equivalent  —JQ,  the  latent  heat 
energy.  In  all  such  cases  dWjdd  must  be  large. 

When  a  gas  expands  isothermally,  there  is  no  change  in  the 
total  internal  energy  /  and  the  change  in  latent  energy  is  equal 
to  the  change  in  free  energy,  or 


.  (102) 

Finally,  in  certain  cases,  the  free  energy  W  is  approximately 
equal  to  the  decrease  in  total  energy  /.  A  stretched  spring 
will  do  an  amount  of  work  approximately  equal  to  the  loss 
in  internal  energy.  The  Daniell  cell  (  §339)  delivers  an  amount 
of  electrical  energy  almost  exactly  equal  to  the  decrease  of 
chemical  energy.  +'" 

The  reader  is  recommended  to  read  the  recent  papers  of 
Nernst  in  which  he  has  attempted  to  find  the  relation  between 
the  change  in  free  energy  (Equilibrium  conditions,  §254)  and  the 
latent  energy  (heat  absorbed).1  It  is  impossible  to  do  more 
than  state  that  his  mathematical  and  experimental  investi- 
gations have  resulted  in  a  general  formula  which  he  has  applied 
to  many  interesting  problems,  such  as  equilibria  at  high 
temperatures. 

Applications  of  the  Second  Law. 

154.  Effect  of  Pressure  upon  the  Freezing  Point  of  Water.— 

Imagine  a  gram  of  water  carried  through  the  following  cycle. 

It  is  allowed  to  freeze  at  o°  and  76  cm.  pressure.     The  air  is 

now  removed  so  that  the  only  pressure  upon  the  ice  is  its  own 

1  These  papers  are  summarized  in  his  "Thermodynamics  and  Chemistry." 


146 


THERMODYNAMICS. 


vapor  pressure  of  .46  cm.  (Table  LVII),  and  at  the  same  time 
it  is  heated  dt°  to  the  melting  point.  It  is  then  melted  at  this 
constant  temperature.  Finally,  the  atmospheric  pressure  is  re- 
stored and  the  temperature  is  lowered  to  zero,  thus  completing 
the  cycle  and  bringing  the  water  back  to  its  initial  state.  The 
cycle  is  illustrated  in  Fig.  61.  Notice  that  the  isothermals  are 


.46- 


FIG.  61. 

in  the  opposite  order  to  what  they  are  in  the  case  of  a  gas, 
i.e.,  the  higher  the  temperature,  the  lower  the  isothermal. 

The  difference  in  the  volumes  of  one  gram  of  water  and  one 
gram  of  ice  is  .0907  c.c.  ^  =  75.54  Xi3-6  XgSo  =  9.52  X  io6. 
Therefore  1^  =  9.52  X. 0907  Xio6.  q,1  the  heat  absorbed,  is 
the  latent  heat  of  fusion,  or  80  calories.  Therefore,  substitut- 
ing in  Equation  100  AM 

Aft 

. 0907X9.5 2  Xio6=  4.187  Xio7X8o  X  — 
J#  =  0.007° 

At  this  temperature,  therefore,  ice  and  water  are  in  equilibrium 
under  the  pressure  of  their  common  vapor. 

155.  Vaporization  and  Sublimation.  Clapeyron's  Equa- 
tion. 

W  =  p(vv-vw) 


q  represents  the  heat  per  gram ;  Q,  the  heat  per  gram  molecule. 


CLAUSIUS'    EQUATION,    TROUTON'S    RULE.  147 

where  vv  is  the  volume  of  the  vapor  and  vw  is  that  of  the  liquid. 
Hence  (Equation  loi)1 

Q  =  -j(vv-vw)^  (103) 

or,  in  the  notation  of  the  calculus 

Q  =  —  j-(vv  —  vw)~Jfi   (Clapeyron's  equation)          (104) 

This  equation  gives  the  relation  between  the  latent  heat  of 
vaporization,  Q  (or  <?!  if  the  volume  of  i  gr.  is  used),  and  the 
rate  of  change  of  vapor  pressure,  dp/dd  at  any  temperature  6. 
Let  us  calculate  by  this  equation  the  latent  heat  of  evapora- 
tion of  water  at  100°.  The  pressure  of  saturated  water  vapor 
is  74.652  cm.  at  99.5°  and  77.369  cm.  at  100.5°. 

Ap 
•'-JQ=  2.717X980X13  59  =  362°8 

vv  =  -  —  (Table  LII)  =  1658  c.c. 

.00060315 

ww=  i  (approximately) 

0  =  373,  /=4.i87Xio7 
Substituting  these  values 
3=  534-5 

which  is  about  2  calories  less  than  the  most  accurate  direct 
experimental  determination  (536.7).2 

156.  Clausius'  Equation,  Trouton's  Rule.  —  The  volume  of 
the  liquid  is  usually  very  small  compared  with  that  of  the 
vapor.  If  we  neglect  it  and  substitute  for  vv  the  value  for  one 
gram  molecule  and  assume  that  the  gas  law  applies  to  the 
vapor  (§107). 

0  RO  dp      RO2   d 


which  is  Clausius'  formula,  which  gives  more  accurate  values 

1  The  change  in  volume  is  neglected  in  comparison  with  the  change  in  pressure. 

2  Winkelmann,  1906,  iii,  p.  1091. 


148  THERMODYNAMICS. 

of  Q  than  Clapeyron's  formula,  if  the  change  in  pressure  is 
large. 

Integrating  Equation  105, 

Inp  —  —  4^  -f  constant 
Ku 

r  7?  n 

.-.  Q  =  0  —  (constant—  Inp)  (106) 

If  we  boil  different  liquids  at  the  same  atmospheric  pressure, 
the  parenthesis  will  have  the  same  value  for  all  liquids,  and 
therefore  the  heat,  Q,  required  to  evaporate  one  gram  molecule 
(=L,  the  molecular  latent  heat  of  vaporization)  is  equal  to 
the  absolute  temperature  of  the  boiling  point  multiplied  by  a 
factor  which  is  the  same  for  all  substances. 

The  values  for  this  constant  for  most  organic  liquids  lie 
between  19.5  and  21.5,  but  water  has  the  value  26  and  acetic 
acid  has  the  extremely  low  value  of  15.  The  deviations  are 
undoubtedly  due  to  uncertainty  regarding  the  actual  molecular 
weight  at  the  boiling  point. 

The  average  experimental  value  of  this  factor  is  2  1  . 

,'.L=2id  (106') 

For  example,  benzol  boils  at  80.2°  and  the  molecular  weight 
13*78.  If  /  represents  the  ordinary  latent  heat  per  gram 


The  experimental  value  of  the  latent  heat  per  gram  is  93. 

PROBLEMS  VI. 

i.  A.  A  body  of  air  which  occupies  1000  c.c.  at  30°,  76  cm.,  expands 
isothermally_to  a  volume  of  2000  c.c.  What  is  (a)  the  final  pres- 
sure? (6)  the"  work  done?  (c)  the  heat  absorbed? 

B.  The  air  then  expands  adiabatically  to  a  volume  of  3000  c.c. 
What  is  (a)  the  final  pressure?  (6)  the  final  temperature?  (c)  the 
work  done? 


3  v* 

\   pdv=p2v2    I    — ,  etc. 


PROBLEMS.  149 

C.  The   air   is   next    compressed   isothermally   from   3000   c.c.    to 
1500   c.c.     Find    (a)  the  final  pressure;  (6)  the  work  done  upon  the 
gas;  (c)  the  heat  emitted. 

D.  Finally,    the    air    is    compressed    adiabatically    to    the    initial 
volume  of   1000  c.c.      Show  that   (a)   the  final  temperature  is  30°, 
(6)  the  final  pressure  is  76  cm. ;  (c)  calculate  the  work  done. 

E.  What  is  the  change  in  entropy  during  (a)  A?    (b]  C? 

F.  What  is  the  theoretical  efficiency  of  the  process? 

G.  Construct  (a)  the  pressure-volume  diagram  of  the  above  cycle; 
(6)  the  entropy-temperature  diagram. 

H.  At  what  rate  does    the   work  in   (a)   A,   (6)   C  vary  with  the 
temperature?      (Equation  102.) 

2.  The  vapor  pressure  of  normal  heptane  is  1851  cm.  at  260°  and 
is    1960    at    264°.     Calculate   the    molecular   latent    heat   per   gram 
(use  Clapeyron's  equation). 

3.  The  vapor  pressure  of  benzol  is  7.5  cm.  at  20°  and  n.8  cm. 
at  30.     Calculate  the  molecular  latent  heat  by  Clausius'  formula. 
(6)    What  is  the  latent  heat  per  gram? 

4.  The  boiling  point  of  methyl  acetate  is  57.1°;  find  by  Trouton's 
law  (a)  the  molecular  latent  heat;  (b)  the  latent  heat  per  gram,      (c) 
What  is  the  latent  heat  of  vaporization  at  20°? 


CHAPTER  III. 
SOLUTIONS.1 

157.  Notation. — The  dissolved  substance  is  called  the  solute, 
the  medium  in  which  it  is  dissolved  is  called  the  solvent,  and 
solvent  and  solute  together  constitute  the  solution. 

158.  Solution  of  Gases,  Henry's  Law. — When  a  gas  and  a 
liquid  are  in  equilibrium,  some  of  the  liquid  has  penetrated 
into  the  gaseous  space  as  vapor,  and  also  some  of  the  gas 
has  dissolved  in  the  liquid.     C.  Henry2  discovered  that  the 
latter  phenomenon  obeyed  the  following  law:     The  mass  of 
gas  dissolved  by  unit  volume  of  the  solvent,  at  a  definite 
temperature,  is  proportional  to  the  pressure  exerted  upon  the 
gas.     Since  the  mass  of  any  volume  of  gas  is  proportional  to 
the  pressure  (Equation  54),  the  volume  of  gas  dissolved  by  unit 
volume  of  liquid  is  independent  of  the  pressure,  if  the  volume 
is  measured  under  this  pressure.     The  absorption  generally 
decreases  with  rise  of  temperature.     More  careful  and  exten- 
sive measurements  since  those  of  Henry  show  that  the  law 
which  bears  his  name  is  only  accurately  true  for  gases  which 
are  "permanent"  (§109,  note),  and  quite  insoluble,  such  as 
oxygen,    nitrogen,    etc.     Gases    which    are    readily   soluble, 
particularly  those  which  give  water  an  acidic  or  basic  nature 
(ammonia,    hydrochloric    acid    gas,    etc.)    show   considerable 
deviation  from  the  law,  particularly  at  high  concentrations. 

159.  The  coefficient  of  absorption  at  zero  degrees  is  defined 
as  the  volume  of  gas  absorbed  by  one  cubic  centimeter  of 
the    liquid    under  a  pressure  of   76   cm.     The  coefficient  of 
absorption  at  any  other  temperature  is  the  volume  absorbed 

1  General    references:      Whetham,     "Theory   of    Solution;"     Jones,    "The 
Modern  Theory  of  Solution;"  Nernst,  "Theoretical  Chemistry;"   Jones,  "Ele- 
ments of  Physical  Chemistry,"  Chap.  V. 

2  Phil.  Trans.,  1803,  i,  p.  29. 


SOLUBILITY    OF    GASES. 


by  one  cubic  centimeter,  but  the  gaseous  volume  is  reduced  to 
zero  degrees. 

TABLE  XIX.  i 

Coefficients  of  Absorption.  (Volume  of  gas,  reduced  to  0°,  absorbed 
by  one  cubic  centimeter  of  liquid  under  a  pressure  of  76  cm.) 


Gas 

Temp. 

Coefficient  of  Absorption  in 
Water  Ethyl  Alcohol 

Nitrogen               .    ... 

0° 
20° 
0° 
20° 
0° 
20° 
0° 
20° 
0° 
20° 
0° 
20° 

.0203 
.014 
.0193 
.0193 
.041  1 
.0284 
.0247 
.0170 
1.797 
.901 
1049.6 
654.0 

.  126 

.120 
.069 
.067 
.284 
.284 

4-33 
2.94 

Hydrogen             

Oxygen 

Air            .             .... 

Carbon  dioxide  
Ammonia  

160.  Solution  of  Gaseous  Mixtures. — If  the  liquid  is  in 
equilibrium  with  a  mixture  of  gases,  their  partial  pressures 
and  total  pressure  are  determined  by  Dalton's  law  (§100), 
and,  therefore,  by  Henry's  law,  the  mass  dissolved  of  each  gas 
is  proportional  to  its  partial  pressure.  This  law  is  obviously 
subject  to  the  deviations  from  both  laws,  which  were  discussed 
in  Chapter  I. 

EXPERIMENT  XIV. 
Solution  of  Gases  in  Liquids. 

The  object  of  this  experiment  is  the  determination  of  the  volume 
of  gas  absorbed  by  a  fixed  volume  of  liquid  and  the  demonstration 
that  this  is  approximately  independent  of  the  pressure.  (Henry's 
Law.) 

The  apparatus  consists  of  a  vertical,  graduated  tube,  with  a  cock  at 
both  ends,  and  connected  at  the  bottom  by  pressure  tubing  with  an  ad- 
justable reservoir.  The  Boyle's  law  apparatus  (Fig.  42)  may  be  used, 
with  minor,  obvious  adaptations,  or  the  special  apparatus  sketched 
in  Fig.  62.  The  apparatus  is  partially  filled  with  mercury.  The 
reservoir  is  raised  until  the  graduated  tube  is  completely  filled  with 
mercury.  The  top  of  the  graduated  tube  is  then  connected  with  the 

1  From  Winkelmann,  1908,  i,  2,  p.  1515. 


SOLUTIONS. 


source  of  gas1  (which  previously  traverses  a  wash-bottle)  and  the 
reservoir  lowered  until  a  volume  of  gas  has  entered  which  occupies  at 
least  half  of  the  graduated  tube.  The  upper  cock  is  then  closed 

and  the  reservoir  adjusted  until  the  gas 
is  under  the  desired  pressure,  which  is 
determined  from  the  difference  in  height 
of  the  two  mercury  columns  and  the 
barometer  reading.  The  level  of  the 
mercury  in  the  graduate  is  carefully 
read.  The  gas  connection  is  now  re- 
placed by  a  funnel  which  is  filled  with 
the  liquid,  care  being  taken  to  remove 
any  air  bubbles  in  the  connections.  The 
reservoir  is  now  lowered  until  the  gas  is 
under  reduced  pressure,  the  upper  stop- 
cock is  opened  slightly  and  a  small 
amount  of  liquid  is  allowed  to  enter, 
after  which  the  cock  is  closed.  The 
volume  of  liquid  is  obtained  from  the 
graduations. 

The  pressure  of  the  gas  is  adjusted  to 
the  original  value  and  the  lower  cock  is 
closed.  The  graduated  tube  is  now 
seized,  and  the  gas  and  liquid  thoroughly 
mixed.  The  lower  cock  is  then  opened, 
and  the  reservoir  is  again  adjusted  until 
the  gas  is  under  the  same  pressure  as  .at 
first.  The  lower  cock  is  then  closed 
again  and  the  mixing,  and  adjustment 
of  pressure  is  repeated,  and  these  steps 
are  continued  until  there  is  no  further 
decrease  of  volume  of  the  gas.  From 
the  level  of  the  liquid  in  the  graduated 
tube  and  the  former  level  of  the  mer 
cury,  the  volume  of  the  gas  absorbed  is 
determined. 

Calculate  the  coefficient  of  absorption 
of  this  particular  gas  and  liquid  at  this 
temperature.  It  is  most  essential  that 
the  temperature  of  the  entire  apparatus 
be  kept  as  constant  as  possible.  Re- 
peat with  the  gas  under  a  different 
pressure. 
FIG.  62. 

QUESTIONS. 

1.  Compare    the    actual    volumes    of    gas    absorbed    at    the   two 
pressures  (i.e.,  not  reduced  to  o°  as  required  by  the  coefficient  of 
absorption). 

2.  Calculate  the  mass  of  gas  dissolved  at  each  pressure  in  one 
litre  of  the  liquid.- 

3.  To  what  error  is  the  experiment  subject? 

4.  Did  the  mercury  levels  remain  stationary  after  the  first  adjust- 
ment ?     How  do  you  explain  the  motion  ? 

1  Carbon  dioxide  is  a  convenient  gas  and  water  or  ethyl  alcohol  is  a  con- 
venient solvent. 


SOLUTION    OF    LIQUIDS    IN    LIQUIDS. 


Solution  of  Liquids  in  Liquids. 

161.  Mutally  Insoluble  Liquids. — Two  liquids  may  be  (A) 
insoluble,  (B)  partially  soluble,  or  (C)  mutally  soluble  in  all 
proportions. 

Water  and  carbon  bisulphide  is  an  approximate  example 
of  (A).  By  Dalton's  law  the  vapor  pressure  above  a  mixture 
of  insoluble  liquids  equals  the  sum  of  the  separate  vapor 
pressures  for  that  temperature.  At  26.87°  the  vapor  pressure 
of  water  is  2.6  cm.  and  that  of  carbon  bisulphide  is  38.9  cm. 
Therefore  the  vapor  pressure  should  be  41.5  cm.  Regnault 
found  the  actual  vapor  pressure  to  be  41.2  cm.  Since  a  liquid 
boils  when  its  vapor  pressure  is  equal  to  the  atmospheric 


51 


88f5 


,105 


100° 


8a°5\. 


_7J>_cirL__  ___/  88°5 


IOO  Water 

0    Isobutyl  Alcohol 


316 


O  Water 
100  Isobutyl  Alcohol 


FIG.  63. 


pressure,  a  mixture  of  two  insoluble  liquids  boils  at  a  tempera- 
ture be.low  the  boiling  point  of  either  of  its  components. 

162.  Partially  Miscible  Liquids. — If  aniline  is  gradually 
added  to  water  at  22°,  solution  takes  place  until  the  concen- 
tration (by  volume)  is  3.48%.  Further  addition  of  aniline 
forms  a  separate  layer  which  dissolves  sufficient  water  from 
the  original  layer  to  form  a  saturated  solution  of  water  (5.22%); 
The  continued  addition  of  aniline  simply  increases  the  amount 
of  this  second  layer,  thereby  reducing  the  first  layer  until  it 
disappears  when  the  volume  of  water  is  but  5.22%  of  that 


SOLUTIONS. 


of   the  aniline.      If  more   aniline  is  added,   an   unsaturated 
solution  of  water  in  aniline  is  formed. 

The  vapors  are  only  slightly  soluble  in  the  liquids  and 
consequently  the  vapor  pressure  of  two  semi-miscible  liquids 
is  usually  greater  and  the  boiling  point  lower  than  that  of 
either  component.  Since  the  concentration  of  neither  layer 
changes  while  both  are  present,  the  vapor  pressure  remains 
constant.  The  heavy  line  of  Fig.  63  represents  the  form  of  an 
isothermal  vapor-pressure  curve  for  two  semi-miscible  liquids 
(water  and  isobutyl  alcohol) .  The  dotted  line  represents  the 
typical  boiling-point  curve. 

TABLE   XX. 

Solubility  of  Semi-miscible  Liquids   (22°). * 


Liquid 

Volume  of  Liquid  in 
100  Volumes  of  Water 

Volume  of  Water  in 
100  Volumes  of  Liquid 

Chloroform  
Carbon  bisulphide  .... 
Ether  

42 
8.1? 

•15 
.96 

2  Q3 

Benzol 

08 

22 

Amyl  alcohol 

3  28 

2  21 

Aniline 

cv-50 

•3  4.8 

522           <• 

Isobutyl  alcohol  

II. 

6. 

163.  Liquids  Miscible  in  all  Proportions. — We  shall  see  later 
that  the  partial  pressure  of  a  solvent  is  less  in  a  solution  than 
in  the  pure  state.  The  vapor  pressure  of  a  solution  of  liquid 
(B)  in  liquid  (A)  is  pA  -f+pB  and  may  be  greater  or  less  than 
pA.  If  the  vapor  of  (B)  is  quite  insoluble  in  (A) ,  pB  more  than 
compensates  for  the  lowering  /.  Water  and  propyl  alcohol  are 
an  example  of  a  mixture  where  both  vapors  are  quite  insoluble. 
The  heavy  line  of  Fig.  64  illustrates  the  general  form  of  iso- 
thermal vapor  pressure  curves  and  the  dotted  line  the  boiling- 
point  curve  for  a  particular  pressure. 

A  solution  with-  the  concentration  represented  by  the  point 
(P)  (e-g->  80%  propyl  alcohol)  has  a  maximum  vapor  pressure 
and  consequently  a  minimum  boiling  point.  Konowalow2 

1  Walker,  p.  53  (except  isobutyl  alcohol). 

2  Wied.  Ann.,  1881,  xiv,  p.  48. 


LIQUIDS    MISCIBLE    IN   ALL    PROPORTIONS. 


has  shown  that  the  vapor  of  a  mixture  with  a  minimum  or 
maximum  boiling  point  has  the  same  composition  as  that  of 
the  liquid.  At  a  lower  concentration  of  alcohol  the  vapor 
has  relatively  more  alcohol  than  the  liquid,  since  propyl  alcohol 
vapor  is  quite  insoluble  in  water.  If  the  amount  of  alcohol 
exceeds  80%  the  vapor  contains  more  of  the  relatively  in- 
soluble water  vapor. 

Therefore,  whatever  the  composition,  the  mixed  vapor  or 
distillate"   approximates  to  the   composition  of  minimum 


100  Water* 
0  Propyl  Alcohol 


80%    o  Water 
100  Propyl  Alcohol 


FIG.  64. 


boiling  point,  while  the  residue  progressively  changes  to 
the  pure  water  or  alcohol,  according  as  the  original  concentra- 
tion is  below  or  above  80%  of  propyl  alcohol.  Noyes  and 
Warfel J  have  shown  that  ethyl  alcohol  has  a  minimum  boiling 
point  at  96°. 

164.  If  each  vapor  is  very  soluble  in  the  other  liquid,  pB  is 
small  and  the  addition  of  either  component  reduces  the  vapor 
pressure  of  the  other.  Formic  acid  and  water  is  an  example 
of  such  a  mixture.  The  vapor  pressure  and  boiling  point 
curves  are  shown  in  Fig.  65.  A  73%  solution  has  a  minimum 
vapor  pressure  and  a  maximum  boiling  point,  and  therefore  by 
Konowalow's  theorem  vapor  and  liquid  have  the  same 
composition.  ^^fb 

1  Am.  Chem.  Soc.,  1901,  xxiii,  p.  463. 


'56 


SOLUTIONS. 


If  the   concentration   of  acid  is  less,    the   vapor  has   less 
acid  than  the  liquid,  owing  to  the  solubility  of  the  vapor.     If 


76 


76  <£}:.  —  " 


100°,.--'-" 


ior 


75.2 


100  Water 
0  Formic  Acid 


73%  0  Water 

100  Formic  Acid 


FIG.  65. 


the  acid  is  more  concentrated,  the  vapor  has  less  water 
than  the  liquid.  Therefore,  whatever  the  concentration, 
continued  boiling  will  give  a  residue  containing  73%  of  acid. 


18.8 


100° 


100  Water 

0  Methyl  Alcohol 


O  Water 
100  Methyl  Alcohol 


FIG.  66. 


nitric  acid  has  a  maximum  boiling  point  at  120°  and 
20.2%  hydrochloric  acid  has  a  maximum  boiling  point  at  1 10°. 


BOILING-POINT   AND    VAPOR-PRESSURE    CURVES.  157 

165.  A  third  type  of  solution  remains,  where  but  one  of  the 
vapors  is  readily  soluble  in  the  other  liquid.  The  solubility 
of  water  vapor  in  methyl  alcohol  is  much  greater  than  that 
of  the  alcohol  vapor  in  water,  and  the  vapor-pressure  curve  for 
65.2°  has  the  form  of  the  heavy  line  of  Fig.  66.  The  dotted 
line  gives  the  corresponding  boiling-point  curve.  The  vapor 
is  therefore  always  richer  in  alcohol,  and  continued  boiling  will 
give  a  residue  of  relatively  pure  water  boiling  at  100°.  By 
successively  reboiling  the  distillate,  practically  all  the  water 
will  be  left  behind  in  the  residues  (fractional  distillation}. 

EXPERIMENT  XV. 

Boiling-point  and  Vapor-pressure  Curves  of  Liquid  Mixtures. 

Determine  the  boiling  points  of  as  many  different  concentrations 
of  water  and  the  assigned  liquid1  as  possible.  Use  the  Beckmann 
boiling-point  apparatus  (§51)  and  start  with  about  20  c.c.  of  the 
assigned  liquid.  When  its  boiling  point  has  been  determined,  add 
i  c.c.  of  distilled  water  from  a  burette  and  again  determine  the  boiling 
point,  and  so  continue.  When  the  total  amount  exceeds  30  c.c.,  pour 
into  a  beaker  and  discard  an  accurately  measured  amount,  so  that, 
on  pouring  the  remainder  into  the  boiling  vessel,  the  amount  is  again 
about  20  c.c.  Continue  until  it  is  almost  pure  water. 

Plot  your  results  with  concentrations  as  abscissae  and  boiling 
temperatures  as  ordinates. 

Make  a  second  diagram  with  concentrations  as  abscissae  and 
vapor  pressures  as  ordinates.  For  each  observed  boiling  point  there 
should  be  a  curve  which  shows  the  general  relation  between  the 
vapor  pressure  and  the  concentration,  at  this  temperature,  i.e.,  an 
isothermal.  In  general  but  three  points  on  each  curve  are  known, 
namely,  the  concentration  at  which  the .  vapor  pressure  equals  the 
atmospheric  pressure,  the  pressure  at  zero  concentration  or  pure 
water  (Table  LVII)  and,  in  certain  cases,  the  pressure  of  the  pure 
solute  (Tables  LIX  and  LX).  Nevertheless,  the  curves  can  be 
drawn  approximately,  for  the  slope  at  each  of  these  points  will  be 
opposite  to  that  of  the  boiling-point  curve,  and  where  the  boiling 
point  is  constant,  the  vapor  pressure  must  be  constant,  and  these 
isothermals  must  be  in  the  order  of  the  temperatures.  It  will  be 
found  convenient  to  draw  a  horizontal  line  at  the  observed  atmospheric 
pressure,  and  to  mark  on  this  line,  at  the  points  corresponding  to 
the  concentrations  studied,  the  temperature  at  which  the  vapor 
pressure  had  this  value  (boiling  points).  Also,  against  the  pressure 
axis,  on  each  side  (pure  solvent  and  pure  solute),  mark  the  pressure 
for  each  of  these  temperatures.  Through  these  points  draw  a  curve 
which  is  horizontal  when  the  boiling-point  curve  is  horizontal  and 
which  slopes  upward  where  the  boiling-point  curve  goes  downward, 

1  Water  and  a  partially  miscible  liquid,  such  as  amyl  alcohol  or  isobutyl 
alcohol,  gives  a  particularly  interesting  and  simple  solution. 


158  SOLUTIONS. 

and  vice  versa.  It  will  be  instructive  to  return  to  this  experiment 
after  Chapter  VII  has  been  finished,  and  interpret  the  curves  in  the 
light  of  the  phase  rule. 

QUESTIONS. 

1.  Explain  why  the  slope  of  the  vapor  pressure  curve  is  opposite 
to  that  of  the  boiling-point  curve. 

2.  If  semi-miscible  liquids  have  been  used,  estimate  (a)  saturated 
concentration  of  each  component ;  (b)  volume  of  each  layer  in  i  oo 
c.c.  of  a  20%  (by  volume)  solution  of  (a)  water;  (b}  liquid. 

1 66.  Solution  of  Solids  in  Liquids. — In  one  respect  such  solu- 
tions are  much  simpler  than  solutions  of  liquids  or  gases  for 
the  vapor  of  the  solute  is  negligible.  The  curves  of  Fig. 
67  illustrate  the  solubility  of  certain  salts  at  different  tempera- 


tures.  The  ordinates  are  the  number  of  grams  of  salt  dis- 
solved in  100  grams  of  water.  The  solubility  of  Glauber's 
salts,  Na2SO4.  10  H2O,  is  particularly  interesting.  Below  34° 
the  solubility  increases  with  rising  temperature,  but  above 
34°  it  decreases.  We  shall  see  in  Chapter  VII  that  at  these 
higher  temperatures  we  are  really  considering  a  different 
salt,  namely  anhydrous  sodium  sulphate. 

167.  Additive  Properties  of  Solutions. — Additive  properties 
are  such  as  can  be  calculated  for  a  solution  or  compound  by 
adding  the  values  of  the  particular  property  for  the  different 


VALSON'S  LAW  OF  MODULI.  159 

constitutents.     If  G  is  the  magnitude  of  a  certain  additive 
property  and  y  the  per  cent,  of  a  certain  constituent, 

ioo£  =  [ioo-  (yl  +  ya  +  -       -)]G0  +  yiGi  +  y2G2  +  - 

(107) 

where  G  refers  to  the  solution,  G0  to  the  solvent,  and  GIt  G2, 
to  the  solutes. 

i67a.  Valson's  Law  of  Moduli. — Valson1  pointed  out  that 
the  density  of  a  salt  solution  is  an  additive  property  and  is 
equal  to  the  sum  of  two  numbers,  one  of  which  is  contributed 
by  the  base  and  the  other  by  the  acid.  If  Za  is  the  number 
for  the  acid  and  Zb  that  for  the  base,  the  density  is 

p=Za  +  Zb 

A  normal  (§4)  solution  of  ammonia  chloride  has  a  low 
density,  1.0153,  and  is  used  as  a  standard.  The  numbers  for 
a  normal  solution  of  any  other  salt,  Za'  and  Zb',  are  deter- 
mined by  the  equation. 

p'  =Za'  +Z&'  =1.0153  +(Za'  -Za)  +(Zbf  -Zb)       (108) 

Zaf  —  Za  is  called  the  modulus  of  the  acid  and  Zb  —  Zb  is 
called  the  modulus  of  the  base.  Table  XXI  gives  the  more 
common  moduli. 

TABLE  XXI. 
Valson's  Moduli  (i8°).» 

NH4  .0000  Cl  .0000 

K  .0296  £SO4  .0160 

Na  -0235  NO  3  .0200 

$Ba  -0739  Br  .0370 

JCa  .0282  I  -0733 

.0221 

.0410 
.0413 
.1090 
Ag  .1069 

The  explanation  of  the  frequent  factor  i  /  2  is  that  such  bases 
or  acids  are  bivalent,  and  therefore  the  equivalent  or  normal 

1  C.  R.,  1871,  Ixxiii,  p.  441;  1873,  Ixxvii,  p.  806. 

2  Reychler-Kuhn,  pp.  120,  121. 


l6o  SOLUTIONS. 

concentration  is  one-half  of  that  calculated  from  the  atomic 
or  molecular  weight.  Table  XXII  gives  the  density  of  dif- 
ferent concentrations  of  ammonium  chloride. 

TABLE  XXII. 

Density  of  Ammonium  Chloride  (18°).     Compiled  from 
Different  Sources. 

Concentration.  Density. 

(gr.  equiv.  per  litre.) 

.5  T.ooSo 

1.  1.0153 

2.  1.0299 

3.  1.0438 
4-  1.0577 

Illustration. — The  density  of  a  thrice-normal  solution  of 
sodium  nitrate  is 

,0  =  1.0438+3  X. 023 5 +3  X- 0200  =  1.1 743 

167!).  Other  Additive  Properties. — Groshaus1  has  shown 
that  the  molecular  volume  (molecular  weight  divided  by  den- 
sity) of  a  salt  solution  is  an  additive  property;  that  is,  is  the 
sum  of  two  terms,  one  of  which  is  contributed  by  the  acid  and 
the  other  by  the  base. 

Von  Bender2  has  calculated  moduli  by  which  the  refrac- 
tive index  of  a  solution  can  be  calculated.  It  is,  however, 
more  logical  to  consider  not  the  refractive  index,  but  the 
molecular  refractive  power  or  refractivity  which  is  defined  by 
the  expression  of  Lorentz-Lorenz.s 

p   ft2  +  2 

where  M/p  is  the  molecular  volume  and  /z  is  the  refractive 
index  (§219). 

If  A  is  the  refractivity  of  a  y  per  cent,  solution,  A0  that  of 
the  solvent,  and  AT  that  of  the  solute 

100  A  =  (100  —  y)  A  ,  +  y  AI  (I09) 

With  many  salt  solutions  the  last  term  can  be  resolved  into 
two,  one  representing  the  refractivity  of  the  acid  and  the 

1  Wied.  Ann.,  1883,  xx,  p.  492. 

2  Wied.  Ann.,  1890,  xxxix,  p.  89. 

3  Wied.  Ann.,  1880,  ix,  p.  641;  xi,  p.  70. 


DENSITY    OF    SALT    SOLUTIONS.  l6l 

other  that  of  the  base,  and  these  terms  possess  approximately 
constant  values,  whatever  the  particular  combination.  The 
refractivity  of  a  solution  will  be  determined  in  Experiment 
XXVII. 

The  power  which  a  salt  solution  possesses  of  absorbing 
light  (§218)  may  be  resolved  into  two  portions,  one  of 
which  depends  on  the  acid  and  the  other  on  the  salt.  For 
example,  all  the  different  permanganates  show  the  same 
characteristic  absorption,  the  explanation  being  that  the 
acid  radical  alone  produces  the  absorption.  Both  the 
natural1  and  the  magnetic2  rotatory  power  (§§226,  228)  are 
additive,  so  that  if,  for  example,  the  acid  radical  is  active, 
the  rotation  is  independent  of  the  particular  inactive  base 
associated  with  it,  and  vice  versa.  Such  cases  will  be  further 
studied  in  Chapter  V. 

PROBLEMS  VII. 

1.  The  space  above  100  c.c.  of  water  (at  20°)  is  filled  with  ammonia 
gas  under  a  pressure  of  200  cm.  of  mercury.     Find  (a)  equivalent 
volume  of  gas  in  water ;  (b)  mass  of  gas  in  water. 

2.  A  litre  vessel  is  filled  with  carbon  dioxide  gas  at  20°  and  76  cm. 
pressure.      How  will  the  pressure  be   affected  by  the  injection  of 
500  c.c.  of  ethyl  alcohol? 

3.  What  is   (a)   the   vapor  pressure   of  a   mixture   of  water  and 
carbon  bisulphide  at  30°?      (Tables  LVII  and  LXI.)      (6)  At  what 
temperature  will  the  mixture  boil  under  a  pressure  of  74   cm.   of 
mercury  ? 

4.  What  is  the  effect  of   continued   boiling  upon  (a)  the  residue 
(6)  the  distillate,  of  70%  ethyl  alcohol?    (c]  98%  ethyl  alcohol? 

5.  Estimate    from    Fig.    67    the    maximum    mass    of    (a)    sodium 
chloride;      (b)    potassium     chlorate;      (c)     sodium    sulphate     which 
will  dissolve  in  one  litre  of  water  at  50°. 

EXPERIMENT  XVI. 
Density  of  Salt  Solutions. 

Prepare  carefully  half-normal  or  normal  solutions  of  several  salts 
having  a  common  base  and  also  similar  solutions  with  a  different  com- 
mon base,  for  example,  i/aNaCl,  i/2NaNO3,  i/4Na2SO4,  i/aNH4Cl, 
i  /2NH4NO3,  i  /4(NH4)2SO4.  Determine  the  densities  with  great  care, 
at  1 8°,  with  a  Mohr  balance  (see  §36  for  full  instructions).  For  each 
acid,  find  the  difference  in  density  for  the  two  bases.  The  difference 
should  be  constant.  Prepare  carefully  similar  solutions  of  two  acids 

1  Oudemann,  Beibl.,  1885,  ix,  p.  635. 

2  Wiedermann,  Magnetismus,  Ladenberg's  Handb.,  vii,  1889. 


162  SOLUTIONS. 

and  different  bases  and  determine  the  densities ;  (for  example,  i  /2  Nad, 
i/2NH4Cl,  i/2KCl,  i/2NaN03,  i/2NH4NO3,  i/2KNO3.  Determine 
the  difference  in  density  contributed  by  these  two  acids  for  each  of 
the  bases.  This  difference  also  should  be  constant. 

Taking  normal  ammonium  chloride  solution  as  a  standard, 
calculate  the  density  moduli.  [If  the  above  solutions  have  been 
used,  the  modulus  of  sodium  will  be  twice  the  difference  between 
the  densities  of  ammonium  chloride  (half-normal)  and  sodium 
chloride  (and  there  should  be  the  same  difference  in  density  between 
each  sodium  salt  and  the  corresponding  ammonium  salt).  The 
modulus  for  the  nitrate  radical  will  be  twice  the  difference  between 
ammonium  nitrate  and  ammonium  chloride  or  any  pair  of  chloride 
and  nitrate  salts  with  a  common  base.] 

QUESTIONS. 

1.  Assuming  i.oi  53  as  the  density  of  a  normal  ammonium  chloride 
solution,    calculate    (a)    the    density   of   a    normal   sodium   chloride 
solution;   (6)   the  density  of  a  twice-normal  solution  of    potassium 
sulphate. 

2.  What  is  the  weight  in  kilos  of  a  cubic  meter  of    half-normal 
sodium  sulphate  solution? 

3.  What   sources  of  error  may  there   be   in   a   determination   of 
density  by  the  Mohr  balance? 

4.  How  might  the  accuracy  of  the  riders  be  tested? 

5.  How  might  the  accuracy  of  graduation  of  the  beam  be  tested? 

6.  What  effect  has  capillarity? 

1 68.  Dissociation.  Association. — The  preceding  para- 
graphs and  experiments  suggest  that  when  a  salt  is  dissolved  in 
water,  it  breaks  up  into  two  parts — an  acid  portion  and  a  base 
portion — and  that  each  of  these  portions  has  characteristic 
properties  which  are  independent  of  the  other  portion.  In 
this  and  in  later  chapters  we  will  have  much  additional  evi- 
dence that  this  is  true  and  that  it  is  not  only  true  of  most  salts, 
but  also  of  bases  and  acids.  We  shall  see  in  Chapter  VIII 
that  the  acid  portion  has  a  charge  of  negative  electricity,  and 
that  hydrogen,  in  the  case  of  acids,  has  a  positive  charge. 
Such  also  has  the  metal,  or  corresponding  radical,  in  the  case 
of  bases  and  salts. 

The  portions  are  called  ions  and  the  general  theory  is 
known  as  the  dissociation  theory  of  Arrhenius.  A  fuller 
discussion  must  be  postponecl  till  later  chapters.  In  certain 
cases  the  molecules,  instead  of  dissociating,  unite  to  form 
groups  of  two  or  more  molecules.  This  phenomenon  is  called 
association. 


OSMOTIC    PRESSURE.  163 

169.  Osmotic  Pressure. — In  1877  Pfeffer1  found  that  when 
a  sugar  solution  is  separated  from  pure  water  by  a  wall  which 
is  permeable  to  water  but  not  to  sugar,  the  water  penetrated 
into  the  solution  until  the  level  of  the  solution  was  consider- 
ably above  that  of  the  pure  water,  or  if  the  solution  rose  into  a 
closed  tube,  the  water  penetrated  until  the  air  was  consider- 
ably compressed  (see  Fig.  69). 

When  the  final  concentration  was  i%  of  cane-sugar, 
this  pressure  was  equivalent  to  50.5  cm.  of  mercury  at 
6.8°,  53.1  cm.  at  14.2°,  54.8  cm.  at  22°,  and  56.7  at  36°. 
Solutions  of  different  concentrations  at  a  common  tempera- 
ture gave  the  following  pressures:  i%  gave  53.5  cm.,  2% 
gave  101.6  cm.,  4%  gave  208.2  cm.,  and  6%  gave  307.5  cm. 
This  pressure  is  called  the  osmotic  pressure  of  the  solution. 

Pfeffer's  experiments  have  been  repeated  by  many  ob- 
servers not  only  with  cane-sugar,  but  with  many  other  solutes, 
such  as  glucose,  manite,  dextrose,  gum  arabic,  and  many  in- 
organic salts.2 

170.  Pfeffer  and  his  successors  used  as  a   semi-permeable 
wall  a  membrane  of  copper  ferrocyanide  which  was  deposited 
in  the  pores  of  a  porous  earthenware  vessel.     This  membrane 
is  permeable  to  water  but  is  quite  impermeable  to  the  above 
solutes.     De  Vries3   found   that   the  walls   of   certain   living 
vegetables  cells  were  in  the  same  sense  semi-permeable.     These 
cells  contained  a  solution,  and  when  surrounded  by  a  similar 
solution  outside,  remained  of  constant  size.     If,  however,  the 
outside  solution  was  weaker,  water  entered  and  the  cell  en- 
larged, and  vice  versa.     He  found  that  it  was  not  necessary 
that  the  outside  solution  be  identical  with  that  inside.     Out- 
side solutions  of  cane-sugar,  glycerine,  etc.,  produced  no  change 
in  the  size  of   the  cell   if   the  molecular  concentration   (§4) 
was  the  same  as  that  of  the  solution  inside.     Such  solutions, 

1  "Osmotische  Untersuchungen,"  Leipsic,  1877,  translated  in  "The  Modern 
Theory  of  Solution,"  Jones. 

2  See  the  numerous  papers  of  Morse  and  his  assistants  in  Am.  Chem.  Jo., 
1909,  xli,  p.   276,  and    previous  numbers;    of    Berkeley    and    Hartley,    Proc. 
Roy.  Soc.,  1904,  Ixxiii,  p.  436;   1906,  Ixxviii,  p.  68,  and  a  general  review  in  Zeit. 
phys.  Chem.  1908,  Ixiv,  p.  i. 

3  Zeit.  phys.  Chem.,  1888,  ii,  p.  415;  1889,  iii,  p.  103. 


164  SOLUTIONS. 

of  equal  osmotic  pressure,  he  called  iso tonic.  De  Vries  found 
that  solutes  which  showed  dissociation,  such  as  the  inorganic 
salts,  sodium  chloride,  potassium  nitrate,  etc.,  were  isotonic 
at  lower  concentrations  than  non-dissociable  solutes,  such  as 
cane-sugar. 

The  osmotic  pressure  of  a  solution,  which  we  have  just  de- 
scribed, appears  to  be  an  actual  pressure  exerted  by  the 
molecules  of  the  solute.  For,  a  non-volatile  substance,  such 
as  cane-sugar,  cannot  leave  the  solution,  and  if  there  were  such 
a  pressure  it  would  be  exerted  on  the  free  surface  of  the 
solution,  on  the  semi-permeable  wall,  and  on  the  other  walls 
of  the  vessel.  Since  the  first  wall  is  permeable  to  the  solvent, 
it  would  experience  no  pressure  from  the  latter,  and  the  other 
walls  are  fixed,  but  the  free  surface  is  not.  Owing  to  this 
pressure  the  volume  of  the  solution  increases  until  this  pres- 
sure is  counteracted  by  the  sum  of  the  pressure  due  to  eleva- 
tion of  the  solution  above  the  pure  solvent  outside,  and  the 
air  pressure  above  the  solution. 

171.  Van't  Hoff J  pointed  out  that  this  pressure  was,  within 
the  limits  of  experimental  errors,  equal  to  the  pressure  which 
would  be  exerted  by  the  dissolved  substance  if  it  occupied 
the  same  volume  in  a  gaseous  state.  Since  the  molecular 
weight  of  cane-sugar  is  342,  a  i%  solution  has  a  molecular 
concentration  of 

i 
34^2 

gram  molecules  per  litre.  By  Equation  60  the  pressure  of  an 
equal  concentration  of  gas  at  14.2°  is 

6235  X  287.2 
1000  p  =  — — - 
34 .2 

•'•  P=  52.4cm. 

while  Pfeffer  found  experimentally  53.1  cm.  pressure.  More- 
over, Van't  Hoff  pointed  out  that  the  pressure  was  proportional 
to  the  absolute  temperature  and  therefore  obeys  Gay  Lussac's 

1  Phil.  Mag.,  1888,  xxvi,  p.  81,  also  "The  Modern  Theory  of  Solution." 


THERMODYNAMIC    STUDY    OF    OSMOTIC    PRESSURE.         165 

law.  The  ratio  of  56.7  to  50.5  is  1.12  while  the  ratio  of 
273+36  to  273+6.8  is  i. ii.  Pfeffer's  figures  quoted  in 
§169  show  that  the  pressure  is  proportional  to  the  concentra- 
tion, that  is,  the  density.  In  other  words,  the  osmotic  pressure 
obeys  Boyle's  law. 

172.  Thermodynamic   Study   of   Osmotic   Pressure. — Van't 
Hoff  not  only  pointed  out  from  Pfeffer's  observations  that 


1% 

I 

V 

C                           A 

B                             D 

FIG.  68. 

osmotic  pressure  followed  Boyle's  and  Gay  Lussac's  laws  and 
iad  the  same  gas  constant,  but  he  showed  that  such  must  be 
the  case  by  considerations  of  an  isothermal  cycle,  involving 
both  gas  pressure  and  osmotic  pressure.1 

The  solute  is  supposed  to  be  such  that,  on  reducing  the  pressure 
above  the  solution,  the  solute  leaves  the  solution  as  a  gas  or  vapor 
and  obeys  Henry's  law  (§158).  The  solution  is  contained  between 
semi-permeable  walls,  AE,  BF,  a  piston  MIf  and  a  free  surface  AB. 
The  walls  AE  and  BF  are  surrounded  on  the  outside  by  the  pure 
)lvent.  The  solute,  when  in  a  gaseous  state,  is  contained  between 

The  following  demonstration  is  due  to  Van't  Hoff,  Rayleigh,  and  Donnan 
(Van't  Hoff,  Lectures  II,  p.  23). 


1 66  SOLUTIONS. 

the  solid  walls  AG  and  BH,  the  piston  M2  and  the  free  surface  A  B. 
The  pressures  p  on  M 2  and  P  on  Mx  have  such  values  that  there  is 
constant  equilibrium. 

We  will  consider  the  following  cycle  of  operations  performed  at 
constant  temperature. 

1.  M2  is  at  A  B.      M±  is  in  a  position  where  the  volume  of  the 
solution  is  V  and  the  pressure  which  must  be  applied  to  this  piston, 
to  counteract  the  osmotic  pressure,  is  P.      Now  raise  M j  to  AB,  while 
M2  is  allowed  to  rise,  against  a  pressure  p,  such  that  the  concen- 
tration of  the  solute  in  the  solution  and  (consequently)  the  osmotic 
pressure    P  remains  constant.     The    work  done  upon  the  solution 
is  — PV  (the  negative  sign  indicating  work  expended).      The  work 
done  by  the  gaseous   solution  is  pv    =  wRO  (Eq.  57)  where  v  is  the 
final  volume  between  A  B  and  M2. 

2.  Allow  the  gas  to  expand  from  v  to  a  very  great  volume  v  y__. 
The  work  done  is  (Eq.  62,  57) 


l/£ 

^x  /» 

I  pdv  =  wRO    I    - 

v  «y 


3.  Lower  piston  Mz  until  the  volume  between  it  and  A  B  is  the 
original  volume  V.     The  volume  above  -AJ3  is  so  great  that  the  con- 
centration of  solute  and  the  osmotic  pressure  below  AB  are  negli- 
gible and  consequently  no  work  is  involved  in  lowering  Mx. 

4.  Lower  M2  to  AB.     The  work  done  upon  the  gas  is 


r     r 

-   I  p'dv=-wRO    I 
o  *J 


dv    --«,ltfln-^ 


v  +  sV  sV 


For  the  solute  not  only  occupies  the  volume  v  above  A  B,  but  also 
the  volume  V  of  the  solution.  We  have  assumed  that  Henry's  law 
is  applicable  and  therefore  a  volume  V  of  the  liquid  absorbs  a  volume 
sV  of  the  gas  above,  where  s  is  the  coefficient  of  solubility  (§159). 
Hence  the  total  equivalent  volume  is  v  +sV.  The  system  has  been 
returned  to  the  initial  state,  and,  since  the  temperature  has  been 
constant,  the  second  law  of  thermodynamics  requires  that  the  total 
work  must  be  zero  (§145) 

v  vM+sV 

.',  -PV  +  wRO -\-WRO\n— --WROln-^-^—^o 
v  sV 

But  sV  =  v  and  this  is  negligible  compared  with  v^ 

.'.PV  =  wR6  =  pv  (no) 

If  the  solute,  as  a  gas,  occupies  the  volume  Vt  the  pressure  p'  is 
determined  by  Boyle's  law. 

p'V  =  pv=PV  .\  p'  =  P  (no') 


VAN'T  HOFF'S  EQUATION  AND  COEFFICIENT.          167 

Therefore,  the  osmotic  pressure  of  a  dissolved,  vaporizable 
substance  which  obeys  Henry's  law  is  equal  to  the  gaseous 
pressure  which  it  would  exhibit  at  the  same  temperature  and 
occupying  the  same  volume.  All  the  deductions  from  this 
law  apply  to  non-vaporizable  solutes,  and  therefore  it  is 
concluded  that  the  law  itself  applies  to  such  solutes  as  well 
as  to  those  which  obey  Henry's  law.  We  have  already  seen 
that  the  experimental  values  for  the  osmotic  pressure  are  in 
agreement.  Equation  57  is  therefore  applicable  to  osmotic 
pressure. 

173.  Van't  Hoff's  Equation  and  Coefficient. — We  have  seen 
that  where  there  is  dissociation  of  molecules  into  two  or  more 
ions,  the  osmotic  pressure  is  greater  than  it  is  if  there  is  no 
dissociation  and  that  the  converse  is  true  of  association.  If 
the  number  of  gram  equivalents  (w  in  Equation  57)  is  calculated 
for  the  normal  molecular  weight,  the  calculated  pressure  will 
be  in  error.  Instead  of  estimating  the  actual  molecular  weight 
it  is  simpler  to  use  the  normal  molecular  weight  and  add  a 
correction  factor.  This  factor  is  called  Van't  Hoff's  coefficient 
and  is  denoted  by  i.  Therefore  the  equation  for  osmotic 
pressure  is 

pv  ==  iRwO  (in) 

If  a  fraction  a  of  the  total  number  of  solute  molecules,  N, 
is  dissociated  into  n  ions,  the  total  number  of  particles  is 
N(i  —a)  +Nna.  Since  i  represents  the  proportional  increase 
in  the  number  of  particles 

.     N  (i  —  a)  +  Nna 


N 


=  i  +  (n—  i)a  (112) 


An  analogous  equation  may  be  worked  out  for  association. 

174.  Diffusion. — If  the  particles  of  a  dissolved  substance 
are  more  concentrated  in  one  portion  of  the  solvent,  the  osmo- 
tic pressure  is  greater,  which  causes  them  to  spread  out  until 
the  concentration  is  uniform.  This  process  is  called  diffusion. 
Pick1  proved  theoretically  and  experimentally  that  the  fol- 

1  Pogg.  Ann.,  1855,  xciv,  p.  59. 


1 68  SOLUTIONS. 

TABLE  XXIII. 

Coefficients  of  Diffusion  (18°).' 
(The  values  below  must  be  divided  by  10?.) 


Concentration 

NaCl 

KC1 

KI 

HC1 

C2H402 

NaOH 

KOH 

•OI                 J35 

169         169 

269 

108           166 

220 

•05 

132     ,    163         163 

261 

104 

1  60 

2I7 

.1 

129         161 

161 

258 

IO2 

158 

215 

•5 

125         156 

159 

253 

99 

i  52         213 

i. 

124         154 

158 

257 

96           149 

215 

lowing  expression  gives  the  quantity  of  salt,  dm,  which  passes 
through  an  area  A  in  time  dT,  when  the  concentration  changes 
an  amount  dc,  in  a  distance  dx,  perpendicular  to  A . 

D  is  a  constant  called  the  coefficient  of  diffusion.  If  we 
interpret  the  equation  in  words,  we  see  that  this  coefficient 
is  the  amount  of  salt  which  would  traverse  one  square  centi- 
meter in  one  second,  if  the  change  of  concentration  per 
centimeter  is  unity.  It  is  customary  to  measure  c  in  grams 
(of  dissolved  substance)  per  cubic  centimeter. 

PROBLEMS  VIII. 

1.  Calculate  the  approximate  osmotic  pressure  of  a  solution  of  i  5 
grams  of  cane-sugar  in  a  litre  of  water  at  (a)  20°  (6)  80°. 

2.  What  is  the  approximate  osmotic  pressure  of  a  solution  of  2 
grams  of  potassium  chloride  in  a  litre  of  water  at  (a)  30°?  (b)  90°, 
assuming  that  the  dissociation  is  .93  in  both  cases? 

3.  If  all  the  molecules  of  benzoic  acid  are  associated  into  double 
molecules  when  5  grams  are  dissolved  in  benzol  to  make  a  solution 
of  50  c.c.,  what  is  the  osmotic  pressure  at  30°? 

4.  50  gr.  of  cane-sugar  are  dissolved  in  a  litre  of  water.      Isotonic 
solutions  are  prepared  of  (a)   ethyl  alcohol,    (b)   hydrochloric  acid. 
Calculate  the  approximate  number  of  grams  of  each  required  per  litre. 

5.  An   11.3%  solution  of  raffinose  is  isotonic  with  a  solution  of 
cane-sugar  containing  19  gr.  mol.  per  litre  (de  Vries).     Calculate  the 
molecular  weight  of  raffinose. 

6.  Derive  an  expression  for  Van't  Hoff's  coefficient  for  a  liquid, 
the  fraction  a'  of  whose  molecules  are  associated  to  double  molecules. 

1  Oholm,  Zeit.  phys.  Chem.,  1904,  1,  p.  309. 


STUDY    OF    OSMOTIC    PRESSURE. 


169 


7.  The  concentration  of  acetic  acid  at  the  bottom  of  a  tank  whose 
cross  section  is  i  square  meter  is  50%.  One  meter  higher  in  the 
tank  the  concentration  is  10%.  How  many  grams  of  acid  will  cross 
any  intermediate  section  in  one  hour  at  18°,  assuming  that  during 
this  time  the  concentrations  do  not  change  appreciably  ? 

EXPERIMENT  XVII. 
Study  of  Osmotic  Pressure. 

The  object  of  this  experiment  is  the  formation  of  a  semi-per- 
meable membrane  which  will  show  qualitatively  the  osmotic  pressure 
of  a  sugar  solution. 


Water 


^ 

'B 

\   C 

c  / 

^> 
B 

\ 

A 

\_ 

L   / 

A 

( 

c 

C 

) 

£ 

1 

i 

) 

5uc 

an 

Solu 

liion 

Water 


A- Cement    B-Paraffm   CrRubber  D=Glass. 
FIG.  69. 

The  semi-permeable  membrane  is  prepared  by  electrolytic  deposi- 
tion of  copper  ferrocyanide  in  the  pores  of  a  porous  cap.1  If  there 

1  Method  of  Morse,  Am.  Chem.  Jo.,  1901,  xxvi,  p.  80.  The  most  suitable 
porous  cups  are  made  by  C.  Desaga,  Heidelberg,  Germany,  of  "Pukalische 
Masse."  Any  unglazed  and  quite  porous  cups  may,  however,  be  used. 


1 70  SOLUTIONS. 

are  remains  of  a  former  membrane,  surround  the  cup,  inside  and  out, 
with  dilute  nitric  acid  for  some  minutes,  after  which  the  cup  should 
be  carefully  washed  with  water.  Mount  a  large  glass  tube  in  the 
top  of  the  cup  by  means  of  a  rubber  collar,  on  top  of  which  is  poured 
a  5:1  cement  of  litharge  and  glycerine.  Coat  the  upper  outside 
portion  with  paraffin.  (See  Fig.  69.)  To  remove  the  air  from  the 
pores,  surround  the  cup  with  a  solution  of  potassium  sulphate  (5  gr. 
per  litre)  and  send  a  current  of  2  amperes  from  a  platinum  electrode 
outside  the  cell  to  a  similar  one  inside,  until  a  large  amount  of  the 
solution  has  passed  through.  Wash  the  cell  carefully.  Pour  inside 
a  tenth-normal  solution  of  potassium  ferrocyanide,  and  outside  a 
tenth-normal  solution  of  copper  sulphate.  Send  a  current,  initially 
of  two  amperes,  from  a  cylindrical  copper  electrode  outside  the  cell 
to  a  platinum  electrode  inside  the  cell,  until  the  formation  of  the 
membrane  practically  stops  the  current.  Be  careful  that  no  potas- 
sium ferrocyanide  comes  outside.  It  may  be  necessary  to  remove 
some  of  the  solution  owing  to  the  electric  endosmosis,  i.e.,  passage 
of  liquid  in  the  direction  of  the  current. 

Rinse  thoroughly  with  water,  fill  with  a  definite  strong  solution 
of  sugar,  and  insert  through  a  rubber  stopper  a  long  capillary  tube, 
being  careful  to  exclude  air  bubbles.  Note  the  pressure  every  ten 
minutes  for  an  hour.  If  the  solution  overflows  the  capillary,  a 
closed  tube  manometer  may  be  attached. 

QUESTIONS. 

1.  Calculate  what  the  pressure  should  be.      (Eq.  m.) 

2.  Why  does  not  the  total  pressure  immediately  manifest  itself? 

3.  Why  is  a  small-bore  tube  preferable  for  measuring  the  pressure? 

4.  What  influence  has  ionization? 

175.  Vapor  Pressure  of  Solutions. — Since  the  free  surface 
of  a  solution  contains  some  molecules  of  solute  and  therefore 
less  molecules  of  solvent  than  the  pure  liquid,  we  would 
naturally  expect  that  the  vapor  pressure  would  be  less.  We 
will  prove  that  such  is  the  case  and  determine  the  magnitude 
of  the  lowering  of  vapor  pressure.  We  will  show  that  if  p 
is  the  vapor  pressure  of  the  pure  solvent  and  p'  is  that  of  the 
solution 

— — —  =  -TJ  =  —  (for  infinitesimal  lowering)      (114) 

where  n  is  the  number  of  gram  molecules  of  solute  dissolved  in 
N  gram  molecules  of  solvent. x 

We  will  imagine  a  very  dilute  solution  contained  in  a  tall  tube, 
with  a  semi-permeable  membrane  at  the  bottom,  which  dips  into  a 
closed  vessel  containing  the  pure  solvent  (Fig.  70). 2 

1  If  the  solvent  has  a  different  molecular  weight  in  liquid  and  vapor,  N  is  the 
number  of  equivalent  gram  molecules  of  vapor. 

2  Demonstration  of  Arrhenius,  Zeit.  phys.  Chem.,  iii,  p.  115. 


VAPOR    PRESSURE    OF    SOLUTIONS. 


171 


Let  h  be  the  equilibrium  difference  of  level  between  the  solution 
and  the  solvent.  Such  equilibrium  must  be  attained,  for  continued 
movement  of  vapor  or  liquid  would  violate  the  conservation  of 
energy.  If  p  is  the  vapor  pressure  of  the  pure  solvent,  the  vapor 
pressure  at  the  level  of  the  solution  is  p—dp,  where  dp  is  the  pressure 
due  to  a  column  of  vapor  of  height,  h.  Since  at  equilibrium  the 


So/Yen  t 


FIG.  70. 

pressure  must  be  the  same  at  all  points  on  the  same  level,  p—dp  is 
the  vapor  pressure  above  the  solution.  The  height  h  therefore 
measures  both  the  lowering  of  the  vapor  pressure  dp  and  the  osmotic 
pressure  P,  and 

dp  mass  of  vapor 


P       mass  of  equal  volume  of  solution 

Let  v  =  volume  of  one  gram  molecule,  M0,  of  vapor.  Let  V  = 
volume  of  solution  containing  one  gram  molecule  of  solute,  pv  = 
PV,  since  the  product  of  pressure  and  volume  for  one  gram  molecule 
of  any  substance,  in  "either  the  vapor  state  or  dissolved,  is  equal 
to  RO  (§172).  Therefore  the  volume  v  which  contains  M0  grams 
of  vapor  is  P/p  times  the  volume  V  of  solution  which  contains  one 
gram  molecule  of  solute.  The  mass  of  V  is  M0N  In,  if  n  gram  mole- 
cules of  solute  are  dissolved  in  N  gram  molecules  of  solvent.  The 
volume  v  of  vapor  is  therefore  P/p  NJn  times  the  volume  of  an  equal 
mass  (M0  gr.)  of  solution.  Hence,  since  the  masses  of  equal  volumes 
are  in  the  inverse  proportion  of  the  volumes  of  equal  masses, 

dp        pn 
P  "  PN 


or 


dp  _    n 
p  ~  N 


(114) 


172  SOLUTIONS. 

If  the  solution  is  moderately  dilute,  we  may  write  p  —  p'  in  place 
dp,  where  p'  is  the  vapor  pressure  over  the  solution, 

Y  ••••*?--*•. •;'•:";       ^ 

Expressed  in  words,  the  relative  lowering  of  vapor  pressure  is 
proportional  to  the  relative  number  of  solute  molecules.  If 
the  change  of  vapor  pressure  is  large,  we  must  integrate 

dp dn_ 

~p  "  ~~N 

between  the  limits  of  pf  and  p;  zero  and  n. 


176.  Illustration.1  —  2.47  grams  of  ethyl  benzoate  dissolved 
in  100  gr.  of  benzol'gave  a  relative  lowering  of  .0123.  Cal- 
culate the  molecular  weight.  Since  the  molecular  weight  of 
benzol  is  78 

N=  —  j--ft;a3  :  .0123  XL  28  =  ^  =  .0158 


..- 

.0158 

Raoult  made  a  great  number  of  observations  upon  the  vapor 
pressures  of  solutions  and  found  close  agreement  with  the 
above  equation.2  The  average  proportional  lowering  for  a 
great  number  of  solutions  for  which  n/N  =  .oi,  was  .0105. 
Measurements  of  the  vapor  pressure  are  difficult  and  therefore 

1  Walker,  p.  179. 

2  Zeit.    phys.    Chem.,    1888,  ii,    p.  372,    also   translated    in  "The  Modern 
Theory  of  Solution." 


ELEVATION    OF    THE    BOILING    POINT. 


it  is  customary  to  measure  the  change  in  boiling  point  or 
melting  point,  both  of  which  depend  upon  the  change  in  vapor 
pressure. 

177.  Elevation  of  the  Boiling  Point. — BC  in  Fig.  71  repre- 
sents the  vapor-pressure  curve  of  the  pure  solvent,  DE  that 
of  the  solution.  If  the  atmospheric  pressure  is  £0,'the  solvent 


At' 


At 


FIG.  71. 


boils  at  the  corresponding  temperature  t.  The  solution  will 
boil  when  its  vapor  pressure  is  equal  to  p0,  which  necessitates 
a  temperature  At  higher.  We  can  easily  observe  this  eleva- 
tion of  the  boiling  point.  Clausius'  equation  (105)  tells  us 

that  •*  -f- ,  /  s  1 


If  m0  is  the  mass  of  the  solvent  and  m  is  the  mass  of  the 
solute,  and  if  M0  is  the  molecular  weight  of  the  solvent  and 
M  that  of  the  solute 


m 


_  At 


M 

mM0 

m0MQ 


174  SOLUTIONS. 

Q/M0  is  the  latent  heat  of  vaporization  per  gram  =•/. 

m 


I       m0M 


(115) 


178.  The  molecular  elevation,  k,  is  denned  as  the  elevation  of 
the  boiling  point  produced  by  dissolving  one  gram  molecule  of 
solute  (w/M  =  i,  §4)  in  100  grams  of  solvent  (w0  =  ioo). 


(116) 


Table  XXIV  gives  the  experimental  molecular  elevations  of 
the  more  common  solvents  at  the  normal  boiling  point.  The 
theoretical  values  of  k  calculated  from  Equation  116  and 
Table  LVI  agree  well  with  these  numbers.  For  example,  for 
benzol  ((?,  (-/  S 

fc==.oi98  (273  +  80.2)*^  26 
93 

The  fourth  column,  which  gives  the  molecular  elevation  for 
100  c.c.  of  solvent,  is  convenient  in  cases  where  the  volume  can 
be  measured  easier  than  the  mass. 

TABLE  XXIV. 

Molecular  Elevations   (Experimental). 


Solvent 

Boil.  Temp. 

Mol.   Elev. 
(100  gr.) 

Mol.   Elev. 
(100   c.c.) 

Acetic  acid 

118 

2  C  2 

26 

*  j-o 

Acetone  

56 

I6.7 

22.2 

Alcohol  

78 

II    ^ 

i  q.6 

Benzol  

80.2 

26.7 

32.8 

Chloroform  

61 

36.6 

2  J. 

Ether    .    .  . 

-2  e 

211 

2  O     "? 

Water 

r    A 

•a 

MOLECULAR    WEIGHT.  175 

From  Equations  115  and  116, 

,,     100  km 


which  gives  the  molecular  weight  of  a  dissolved  substance  in 
terms  of  easily  observed  quantities  and  the  constant  of  the 
solvent. 

EXPERIMENT  XVIII. 

Determination  of  Molecular  Weight  of  a  Dissolved  Substance  from 
the  Elevation  of  the  Boiling  Point. 

(Beckmann  Apparatus.) 

Study  carefully  the  directions  for  the  boiling-point  apparatus 
(§51)  and  the  Beckmann  thermometer  (§47).  Having  several 
times  determined  the  boiling  point  of  the  pure  solvent,  add  the 
assigned1  solute  in  successive  quantities  of  from  0.2  to  0.6  grams, 
depending  on  its  solubility  and  molecular  weight,  and  for  each 
addition  determine  the  boiling  point.  Solid  substances  should  be 
pressed  into  convenient  pellets.  Liquids  should  be  introduced 
from  a  special  pipette  with  a  long  bent  neck,  and  of  such  a  form 
that  it  can  be  hung  in  a  balance,  and  the  liquid  expelled  determined 
by  weight  (§36).  Calculate  the  molecular  weight  for  each  concen- 
tration, by  Equation  117.  Compare  the  result  with  that  represented 
by  the  chemical  formula. 

QUESTIONS. 

1.  Explain  any  abnormal  molecular  weights  observed. 

2.  Why  must  the  bulb  of  the  thermometer  be  in  the  liquid  rather 
than  in  the  vapor? 

3.  Calculate  the  boiling  point  of  a  solution  whose  concentration 
is   (a)    i    gram   of  the   assigned  solute  in  200  gr.  of  the  solvent  (b) 
one  gram  molecule  of  the  solute  in  too  gram  molecules  of  the  solvent. 

4.  Calculate  the  molecular  elevation  of  this  solvent  (Equation  116 
and  Table  LVI). 

EXPERIMENT  XIX. 

Determination  of  the  Molecular  Weight  of   a  Dissolved  Substance  by 
Elevation  of  the  Boiling  Point. 
(Landesberger-Walker  Method.) 

Before  beginning  the  experiment,  study  carefully  the  directions  for 
the  use  of  this  apparatus  (§52)  and  the  Beckmann  thermometer  (§47). 
Make  several  observations  of  the  boiling  point  of  the  pure  solvent  and 

1  Any  one  of  the  solvents  in  Table  XXIV  is  suitable.  The  solute  may  be  any 
non-volatile  or  slightly  volatile  substance  of  definite  chemical  composition.  A 
substance  which  shows  dissociation  (e.g.,  an  inorganic  salt),  or  one  showing 
association  (e.g.,  benzoic  acid  in  benzol)  is  particularly  interesting. 


176  SOLUTIONS. 

then  observations  with  several  concentrations  of  the  assigned  solute.1 
Solid  solutes  are  preferably  pressed  into  small  pellets,  and  liquid 
solutes  should  be  discharged  from  a  weighing  pipette  (§36).  Calculate 
the  molecular  weight  for  each  concentration  by  Equation  117,  using 
the  value  of  k  for  100  c.c.  (Table  XXIV),  and  compare  the  results 
with  that  represented  by  the  chemical  formula 

QUESTIONS. 

1.  Explain  any  abnormal  molecular  weights  obtained. 

2.  What   advantages  and  what   disadvantages   has  this  method 
compared  with  the  Beckmann  method? 

3.  Calculate  the  boiling  point   of  a  solution  containing    (a)    one 
gram  of  this  solute  in  100  grams  of  this  solvent ;  (b)  one  gram  molecule 
of  this  solute  in  100  gram  molecules  of  this  solvent. 

4.  Calculate   the   molecular    elevation  of   this   solvent  (Equation 
116  and  Table  LVI). 

5.  Reconcile  this  transfer  of  heat  to  the  warmer  solution  with  the 
second  law  of  thermodynamics. 

179.  Lowering  of  the  Freezing  Point. — We  shall  see  later 
that  when  a  solution  begins  to  solidify,  the  solid  pure  solvent 
separates  out  and  the  vapor  pressure  above  the  solution  is 
equal  to  that  above  the  solid  pure  solvent  at  this  temperature. 
If,  therefore,  AB  (Fig.  71)  represents  the  vapor-pressure  curve 
of  the  solid  solvent,  the  solution  begins  to  freeze  at  the  tempera- 
ture corresponding  to  the  point  D  where  the  vapor-pressure 
curve  of  the  solution  meets  that  of  the  solid  solvent.  At', 
the  lowering  of  the  freezing  point  by  the  addition  of  the  solute, 
can  be  expressed  in  terms  of  Ap',  the  lowering  of  vapor  pressure 
of  the  solid  solvent. 

Let  m0=mass  of  solvent,  w=mass  of  solute,  and  M=mo- 
lecular  weight  of  solute.  The  number  of  gram  molecules  of 

solute  is  j-=-  and  consequently  one  gram  molecule  of  solute  is 

contained  in  — - —  grams  of  solvent.     Let  V=  volume  of  this 
m 

mass  of  solvent.     If  this  amount  of  solvent  freezes  out  from 

the  solution,  the  latent  heat  liberated  is  /' —    -  and  by  §148 

m 

At' 

the  fraction  -^-  of  this  heat  does  the  mechanical  work  neces- 
v 

1  Ethyl  alcohol  and  ether  are  the  best  solvents.  The  solute  may  be  any 
substance  of  definite  chemical  composition  which  is  readily  soluble  in  either  of 
these  solvents  and  which  is  only  slightly  volatile. 


MOLECULAR  LOWERINGS. 


I77 


sary  to  reduce  the  volume  of  the  solution  by  an  amount  V 
against  the  osmotic  pressure  P.     Therefore, 


TJf  "  *&•'•*• 

JL~™r~o~= 

and  the  latter,  by  §172,  is  equal  to  RO.     Therefore, 

f_Rd2     m 
'~~JV~mM 

TABLE  XXV. 
Molecular  Lowerings  (Experimental). 


(118) 


Solvent 

Freezing  Point 

Molecular  Lowering 
(100  gr.) 

Water 

o 

18  =; 

Benzol 

5     A 

TO 

Nitrobenzol 

7O  7 

Acetic  acid 

I  7 

18  6 

180.  The  molecular  lowering  k'  is  denned  as  the  lowering 
of  the  freezing  point  produced  by  dissolving  one  gram  mole- 
cule of  solute 


in  100  grams  of  solvent  (w0  =  ioo) 

.0198^ 
~^T 

which  is  an  analogous  expression  to  Equation  116.  For  ex- 
ample, for  benzol 

_  .0198(273  +  5.4)*  _ 

K     -  —   §  i 

32.2 

Table  XXV  gives  the  experimental  values  of  the  molecular 
lowering  of  the  more  common  solvents.  The  figures  agree 
well  with  the  calculated  values. 


178  SOLUTIONS. 

From  Equations  118  and  119 

,,      look'm 

M=^f^r  (I20) 

With  the  help  of  this  equation,  the  molecular  weight  of 
a  dissolved  substance  can  be  determined  from  easily  observed 
quantities  and  the  constant  of  the  solvent.  The  freezing-point 
is  preferable  to  the  boiling-point  method  because  it  is  inde- 
pendent of  the  vapor  pressure  of  the  solute. 

181.  Molecular  Weight   in   Solution. — Equations    117    and 
120  are  the  bases  of  the  two  best  methods  for  determining  the 
molecular  weights  of  bodies  in  solution.     It  is  only  necessary 
to  know  the  mass  of  solvent  m0,  the  mass  m  of  the  substance, 
the  difference  between  either  the  boiling  point  or  the  freezing 
point  and  that  of  the  pure  solvent,  and  the  constant  of  the 
solvent  (Tables  XXIV  and  XXV).     If  the  molecular  weight 
in  solution  is  lower  than  the  normal  molecular  weight,   it 
shows  that  the  molecules  have  partially  or  wholly  dissociated. 
Solution  of  salts,  acids,  and  bases  show  dissociation,  and  the 
dissociation,  as  measured  by  the  decrease  in  molecular  weight, 
increases  if  the  solution  is  diluted  (compare  Table  XXXVI) . 
An  abnormally  high  molecular  weight,  on  the  other  hand,  shows 
that  there  is  association  of  the  molecules  into  groups  of  two 
or  more.     The  above  formulae  are  inapplicable  if  the  molec- 
ular weight  of  the  solvent  is  different  in  liquid  and  vapor,  and 
Equation  117  cannot  be  used  if  the  solvent  is  volatile  (§175 
and  note) . 

182.  Solid  Solutions. — Solids  as  well  as  liquids  often  form 
what  may  be  very  properly  called  solutions.     For  example, 
palladium   dissolves   hydrogen  gas,  and   the   various   alloys, 
in  particular  steel,   are  mixtures  which  show  many  of  the 
properties    of    ordinary    solutions.     A    discussion    of    these 
solutions  must,  however,  be  postponed  until  after  we  have 
considered  the  phase  rule.1 

183.  Colloidal  Solutions. — A  colloidal  solution  is  one  where 
there  is  (usually)  great  association  (§168)  of  the  solute  particles. 

1  For  detailed  information  respecting  solid  solution,  see  the  Lectures  of 
Van't  Hoff,  the  pioneer  in  such  investigations  (Part  I,  i,  §5;  Part  II,  i,  §3). 


COLLOIDAL  SOLUTIONS.  179 

A  solution  of  gelatine  in  water  is  a  reversible  colloid  solution; 
that  is,  if,  by  drying  or  cooling,  it  forms  a  solid  jelly,  heating 
or  the  addition  of  water  will  restore  it  to  solution. 

Other  colloidal  solutions,  such  as  that  of  ferric  hydroxide, 
or  finely  divided  metals  will  not  go  into  solution  if  once 
coagulated.  Such  irreversible  colloids  appear  to  carry  electric 
charges,  and  are  coagulated  by  the  addition  of  a  highly  ionized 
solute.  The  reader  is  referred  to  the  references  for  fuller 
information.1 

PROBLEMS  IX. 

1 .  Pfeffer  found  that  a  i  %  colloidal  solution  of  dextrine  gave  an 
osmotic  pressure  of  16.6  cm.  at  16°.     Calculate  the  molecular  weight. 

2.  What  is  the  relative  lowering  of  vapor  pressure  produced  by 
dissolving  (a)  i  5  grams  of  cane-sugar  in  a  litre  of  water?     (b)  2  grams 
of  potassium  chloride?     (c)   What    is   the  actual    vapor  pressure  of 
each  solution  at  ioo°? 

3.  Calculate  the  boiling  point  of  both  solutions  of  Problem  2. 

4.  Calculate  the  freezing  point  of  both  solutions  of  Problem  2. 

5.  Calculate  for  water  (a)  the  molecular  elevation;  (b)  the  molecular 
lowering. 

6.  If  i  gr.  of  benzoic  acid  lowers  the  freezing  point  of  20  c.c.  of 
water  .24 ;  of  20  c.c.  of  benzol  .67,  and  of  20  c.c.  of  acetone  i,  calculate 
the  molecular  weight  in  each.     Explain  the  differences. 

EXPERIMENT  XX. 

Determination  of  the  Molecular  Weight  of  a  Dissolved  Substance  by 
Lowering  of  the  Melting  Point. 

Study  carefully  the  directions  in  §50  for  the  use  of  the  Beck- 
mann  apparatus  and  §47  for  the  adjustments  of  the  Beckmann 
thermometer. 

Having  made  several  careful  determinations  of  the  freezing  point 
of  the  pure  solvent,2  add  the  assigned  solute  in  successive  amounts  of 
from  .2  to  .6  grams,  depending  upon  the  molecular  weight,  and  for 
each  concentration  make  a  careful  determination  of  the  freezing 
point.  Solid  solutes  are  preferably  pressed  into  pellets  and  liquid 
solutes  should  be  discharged  from  a  weighing  pipette  (§36). 

Calculate  the  molecular  weight  for  each  concentration  by  Equation 
120  and  compare  the  results  with  that  represented  by  the  chemical 
formula. 

'Wood  and  Hardy,  Pro.  Roy.  Soc.,  1909,  Ixxxi,  B,  545,  p.  38;'  Burton 
Phil.  Mag.,  1906,  xi,  p.  425. 

2  Any  of  the  solvents  of  Table  XXV  are  suitable,  and  the  solute  may  be  any 
readily  soluble  substance  of  definite  chemical  composition.  Acetic  acid  requires 
special  precautions  on  account  of  its  hygroscopic  nature.  It  is  particularly 
interesting  to  use  a  solute  which  shows  dissociation  or  association. 


i8o 


SOLUTIONS. 


QUESTIONS. 


1.  Explain  why  the  freezing  point  does  not  remain  constant  during 
the  freezing  of  a  solution. 

2.  If  the  freezing  point  of  acetic  acid  used  as  a  solvent  is  higher 
than  that  of  pure  acetic  acid,  how  could  you  determine  the  amount 
of  water  present  ? 

3.  How  could  you  determine  which  of  two  specimens  of  a  liquid 
is  the  purer? 

4.  Calculate    the    coefficient    of    lowering    for    the    solvent    used. 
(Equation  119  and  Table  LVI.) 

5.  Explain  any  abnormal  molecular  weights. 

6.  Calculate  the  approximate  freezing  point  of  a  solution  of  this 
solute  in  this  solvent,  the  concentration  being  (a)  one  gram  in  100 
grams  of  solvent,   (6)  one  gram  molecule  in  100  gram  molecules  of 
solvent. 


CHAPTER  IV. 
THERMOCHEMISTRY.1 

184.  Heat  of  Formation. — If  two  bodies  unite  to  form  a 
chemical  compound  or  a  physical  solution,  the  energy  of  the 
united  bodies  is  generally  different  from  the  sum  of  the  energies 
of  the  constituents.  For  example,  at  a  moderate  temperature, 
hydrogen  and  oxygen  unite  to  form  water.  For  each  gram- 
molecule  of  water  vapor  formed,  58,700  calories  of  heat  energy 
are  given  to  surrounding  bodies.  Therefore,  the  energy  of  a 
gram  molecule  of  water  vapor  must  be  about  58,700  calories 
less  than  that  of  a  grammolecule  of  hydrogen  plus  that  of 
half  a  grammolecule  of  oxygen.  We  can  state  the  reacting 
substances,  the  final  product,-  and  the  energy  change  as 

H2  +  iO.  =  H20+  58700 
or,  in  Thomsen's  notation, 

(H2,0)  =  58700 

This  energy  is  generally  in  the  form  of  heat  energy,  but 
it  may  take  the  form  of  electrical  energy  (§321)  or  mechanical 
work  (§129).  For  example, 

Zn  +H2SO4-ag  =  ZnSO4'ag  +  H2  +  248000 2 

248,000  calories  do  not  represent  the  entire  loss  of  energy, 
for  some  energy  is  consumed  by  the  hydrogen  gas  in  over- 
coming the  outside  atmosphere.  This  work  is  equal  to 

P(v2~v,)=pv2  =  RO  (Eq.sy)  (121) 

for  the  volume  vt  of  the  constituents  is  negligible  compared 
with  the  volume  v2  of  the  gas.     If  the  temperature  is  27°, 

1  General  references  for  Chapter  IV:  Thomsen,  "Thermo  Chemistry,"  trans- 
lated  by    Burke;    Nernst,    "Theoretical    Chemistry,"    Book   IV;    Reychler- 
Kiihn,  Part  III,  i. 

2  The  symbol  aq  signifies  that  the  substance  is  in  dilute  solution. 

181 


182  THERMOCHEMISTRY. 

^#  =  1.985X300  =  595  (Eq.  59)  and  the  total  energy  change  is 
248,600  calories.  If  the  zinc  is  an  element  of  a  galvanic  cell, 
no  hydrogen  is  evolved  and  248,600  calories  or  1,040,000  joules 
of  electrical  energy  are  produced.1  In  general, 

I=W-Q  (122) 

where  —  Q  is  the  heat  emitted,  W  is  the  external  work  done 
and  /  is  the  decrease  in  internal  energy  (Equation  78).  This 
decrease  in  the  internal  energy  is  called  the  heat  of  formation, 
or,  better,  the  heat  accompanying  the  formation  of  the  new  sub- 
stance or  substances. 

185.  In  many  cases  I  can  be  measured  without  difficulty  by 
causing  the   entire   energy  change   to   appear  as  heat,   and 
measuring  this  heat  in  a  suitable  calorimeter  (§  §53-58).     The 
result  will  evidently  be  different  according  as  the  union  takes 
place  at  constant  pressure  or  constant  volume.     For  example, 
in  the  illustration  of  the  formation  of  water,   the  number 
of  molecules,  and  therefore  the  volume,  is  decreased  one-third. 
If,    therefore,    the   reaction   proceeds   at    constant   pressure, 
the    external    atmosphere  does    an  amount  of   work   i/^RO 
which  it  does  not  do  when  the  reaction  proceeds  at  constant 
volume,  and  therefore  the  heat  produced  is  greater  by  this 
amount  when  the  pressure  rather  than  the  volume  is  constant. 
Unless  otherwise  specified,  we  will  understand  that  the  heat 
of  formation  is  measured  at  constant  pressure  or  reduced  to 
such  a  condition,  and  also  that  before  a  final  estimate  of  the 
heat,  the  final  products  are  brought  to  the  initial  temperature 
of  the  original  bodies. 

186.  Heat  of  Solution. — Physical  examples  of  the  energy 
change  accompanying  the  union  of  two  bodies  are  the  so-called 
heats  of  solution  and  dilution.     The  heat  of  solution  is  the 
number  of  calories  of  heat  emitted  during  the  solution  of  one 
gram  molecule  of  the  substance  in  so  great  an  amount  of  water 
that   further   addition   of   water   causes   no   additional   heat 
emission  or  absorption.     The  heat  of  solution  of  anhydrous 
calcium  nitrate  is  4000  calories;  that  is,   when   164  gr.   are 

1  The  heat  of  formation  of  hydrogen  ions  is  negligible  (§201). 


HEAT    OF    DILUTION.  183 

dissolved  in  a  large  amount  of  water,  4000  calories  of  heat 
energy  are  emitted.  The  heat  of  solution  of  hydrates  is  less 
than  that  of  the  anhydrous  salts.  For  example,  the  hydrate 
of  calcium  nitrate,  Ca(NO3)24H2O,  has  a  heat  of  solution  of 
—  7600.  (The  negation  sign  means  that  the  solution  is  ac- 
companied by  an  absorption  of  heat.) 

187.  Heat  of  Dilution. — The  heat  of  dilution  of  a  solution  is 
the  amount  of  heat  per  gram  molecule  of  dissolved  substance 
which  is  emitted  when  the  solution  is  greatly  diluted.  Beyond 
a  certain  dilution,  further  addition  of  water  produces  no  ap- 
preciable affect.  While  a  particular  solvent  and  a  particular 
dissolved  substance  have  a  definite  heat  of  solution,  they  have 
no  definite  heat  of  dilution,  since  the  latter  will  of  course 
decrease  as  the  dilution  becomes  greater.  Table  XXVII 
illustrates  this  fact. 

Practically  the  only  substances  without  appreciable  heats 
of  solution  or  dilution  are  gases  which  closely  obey  Henry's 
law  (§158).  The  general  existence  of  such  heats  is  an 
argument  of  Kahlenberg  and  others  that  solution  is  akin 
to  chemical  combination  rather  than  a  purely  physical 
phenomenon. 

The  heat  of  formation  of  the  more  common  compounds  and 
their  heats  of  solution  in  water  are  given  in  Table  XXVI. 
To  avoid  long  numbers,  the  values  given  are  in  large  calories 
(§7).  To  reduce  to  the  common  calorie,  multiply  the  values 
given  by  1000. 

TABLE  XXVII. 

Heat  of  Dilution  of  Different   Solutions  of  Nitric  Acid.1 
(Heat  of  Solution  =  71 50.) 

HN03  +  H20 3840 

+  2H2O 2320 

+  3H2O 1420 

+  4H2O 790 

+  6H2O 200 

+  8H2O -40 

•f-iooH2O -30 


1 84 


THERMOCHEMISTRY. 


TABLE  XXVI. 
Heats  of  Formation  and  Heats  of  Solution  (18°).* 

(The  unit  is  the  large  calorie  =  1000  common  calories.  The  nega- 
tive sign  signifies  that  heat  is  absorbed  during  the  formation  of  the 
compound.) 


Compound 

Chemical 
Symbol 

Heat  of 
Formation 

Heat  of 
Solution 

Ozone                       

O3 

—  34-  i  2 

Water   vapor          \ 

f  58.7 

Water,  liquid    / 
Hydrogen  dioxide  

H2O 
H2O2 

168.4 
45  •  2 

Hydrochloric  acid 

HC1 

22  . 

20  .  3 

Hydrobromic  acid     
(gaseous  bromine) 
Hydriodic  acid 

HBr 
HI 

12  .  I 

—6.  i 

19.9 

19  .  2 

(solid  iodine) 
Hydrogen  sulphide        .... 

H2S 

2  .  7 

4.6 

Sulphuric  acid                

H2SO4 

IQT,  .  I 

17.8 

Ammonia   
Nitric  acid  

NH3 
HNO3 

12  • 

41  .  9 

8.4 

7  .  2 

Nitrous  oxide  

N2O 

—  18. 

Nitric  oxide   

NO 

—21.6 

Nitrogen  peroxide  

/N204 

—  2.6 

Nitrogen  pentoxide  
Phosphoric  acid 

\  NO2 
N20< 
H,PO4 

—7-7 
J3-1 

T.O2  .  Q 

2  .  7 

Carbon  dioxide 

c62 

07-6 

6.0 

(from  amorphous  carbon) 
Carbon  monoxide  

CO 

2Q  . 

Methane   

CH4 

22-4 

Carbon  tetrachloride    
Carbon  bisulphide  
Hydrocyanic  acid   

CC14 
CS2 
HCN 

21.6 

IQ- 
—  27.6 

Potassium  hydroxide  

KOH 

IO3  .  2 

13  .3 

Potassium  chloride    
Potassium  chlorate 

KC1 
KC1O3 

104.3 

Q  C  .  Q 

—  3-1 

Potassium  perchlorate  . 

KC1O4 

I  I  T.   .   I 

12  .  I 

Potassium  bromide  

KBr 

Q  s;.  i 

—  5  •  J 

Potassium  iodide     

KI 

80.  i 

—5.  i 

Potassium  sulphate     
Potassium  nitrate   
Potassium  sulphide 

K2S04 
KN03 

K2S 

344-6 
119.5 

IOI  .  2 

—6.4 
-8.5 
10  .0 

Potassium  carbonate    
Potassium  permanganate     .  .  . 
Sodium  hydroxide    
Sodium  chloride 

K2C03 
KMnO4 
:   NaOH 

NaCl 

28l  .  I 

!95- 
ioi  .  9 

07     6 

-6-5 
—  10.4 
10.  9 

I  .  2 

Sodium  bromide 

NaBr 

St.  8 

.  2 

1  Compiled  from  the  excellent  table  in  Juptner,  vol.  i,  Chap.  XXII. 

2  Jahn,  Zeit.  anorg.  Chem.,  1908,  Ix,  3,  p.  292. 


HEATS    OF    FORMATION   AND    SOLUTION. 
TABLE  XXVI.— Continued. 


Compound 

Chemical 
Symbol 

Heat  of 
Formation 

Heat  of 
Solution 

Sodium  sulphide    .         

Na2S 

87  .0 

I  C  . 

Sodium  hyposulphite    

Na2S2O3-5aq 

26^.2 

—  1  1  .  4 

Sodium  sulphite    . 

Na2SO3 

268.  5 

—  1  1  .  i 

Sodium  sulphate 

Na2SO4 

328.8 

.  2 

Sodium  nitrate      

NaNO3 

I  I  I  .  ^ 

—  *—G  . 

Sodium  carbonate 

Na2CO3 

272.6 

c.6 

Ammonium  chloride 

NH4C1 

7^.8 

—  4  • 

Ammonium  sulphate    . 

(NH4)2SO4 

282  .2 

—2  .6 

Ammonium  nitrate  . 

NH4NCK 

88. 

—6.2 

Calcium  hydroxide    
Calcium  oxide 

gSOH)* 

215. 

1  1.  1  . 

(Heat    of 

Calcium  chloride     
Calcium  carbonate    
Magnesium  hydroxide 

CaCl2 
CaCO3 
Mg(OH)2 

170. 
270. 

217.3 

hydration 

=  iS-5) 
17.4 

Magnesium  sulphate 

MgSO4 

502  . 

2O  .  3 

Aluminum  hydroxide  
Manganese  hydroxide 

A1(OH)3 
Mn(OH), 

297. 
163  . 

Ferrous  hydroxide    
Ferrous  chloride 

Fe(OH)2 
FeCl2 

136.7 
82 

I7.O 

Ferrous  sulphate     
Ferric  hydroxide      
Ferric  chloride  
Cobalt  hydroxide    

FeSO4'aq 
Fe(OH)3 
FeCl3 
Co(OH)2 

235-6 
198. 
96.  i 
131  .8 

63.3 

Cobalt  chloride 

CoCl2 

76    <; 

18.3 

Nickel  hydroxide     
Nickel  chloride 

Ni(OH)2 
NiCl2 

/    o 

129.2 

74.  .  ^ 

6 
IQ  .  2 

Zinc  oxide    . 

ZnO 

85  8 

Zinc  chloride. 

ZnCl2 

07  . 

i  s.6 

Zinc  sulphide 

ZnS  'aq 

"?  o   6 

Zinc  sulphate 

ZnSO4 

220. 

18.  c 

Cadmium  hydroxide  

Cd(OH)2 

i  34  .  i 

Cadmium  chloride 

CdCl2 

03     2 

3  . 

Cupric  oxide 

CuO 

77     2 

Cupric  chloride 

CuCl2 

51.6 

1  1  .  I 

Cupric  sulphate 

CuSO4 

182  .  6 

15.8 

Cupric  nitrate    . 

CufNOOa'aq 

82.3 

Cuprous  oxide   . 

Cu2O 

40  .  8 

Cuprous  chloride  . 

Cu2O 

40  .  8 

Cuprous  chloride     . 

Cu2Cl2 

6s.  7 

Mercurous  oxide  
Mercurous  chloride 

Hg20 
Hg2Cl2 

22  .2 

62  .6 

Mercuric  oxide  
Mercuric  chloride  ... 

HgO 
HgCl2 

20.7 

C3  .  2 

—  3  .  3 

Potassium  amalgam 

KHg12 

34  . 

Sodium  amalgam    .  . 

NaHg6 

2  I  .  I 

Silver  oxide  

Ag2O 

C  .  Q 

i86 


THERMOCHEMISTRY. 
TABLE  XXVI  -  Continued. 


Compound 

Chemical 
Symbol 

Heat  of 
Formation 

Heat  of 
Solution 

Silver  chloride 

AgCl 

20.4 

Silver  bromide  
Silver  iodide     
Silver  nitrate     

AgBr 
Agl 
AeNO, 

22.7 
13.8 
28.7 

—  5  -4 

Stannous  chloride     
Stannic  chloride    ...... 

SnCl,  3 
SnCl4 

80.8 
127.3 

•3 
29  .  9 

Lead  oxide  
Lead  chloride     

PbO 

PbCl2 

50-3 
82.8 

—6.8 

Lead  sulphate 

PbSO4 

216.2 

Lead  nitrate     

Pb(NO3)2 

105  .  5 

—7.6 

188.  The  Principle  of  Stable  Equilibrium. — It  is  a  funda- 
mental  principle   of  mechanics  that   a  body  is  in  a   stable 
equilibrium  when  its  potential  energy  is  a  minimum.     When 
such  is  the  case,  any  disturbance  increases  the  potential  energy 
and  thereby  assists  the  return  to  the  initial  condition. 

Berthelot2  in  1879  announced  that  an  analogous  principle 
was  applicable  to  chemistry.  We  have  seen  that  the  tempera- 
ture is  a  measure  of  the  kinetic  energy  of  the  molecules 
(§104).  An  increase  of  temperature  or  an  emission  of  heat 
would  usually  mean,  therefore,  a  decrease  of  potential  energy, 
and  therefore  it  would  be  expected  that  most  reactions  would 
be  of  this  type. 

189.  The  Principle  of  Maximum  Work. — Berthelot's  theo- 
rem states  that  every  spontaneous  chemical  transformation 
proceeds  in  the  manner  which  will  be  accompanied  by  the 
greatest    emission    of    heat.     Table    XXVI    illustrates    the 
general    applicability    of   this    theorem,  for  the    compounds 
are  stable  and,  with  a  few  exceptions,  the  heats  of  formation 
are   positive.     These   exceptions,    ozone,    oxide    of   nitrogen, 
carbon  bisulphide,  etc.,  together  with  the  majority  of  heats 
of  solution,  show  that  this  theorem  is  not  a  natural  law. 

190.  Le  Chatelier's  Principle. — Le  Chatelier  greatly  restricted 

1  Reychler-Kiihn,  p.  166. 

2  Essai  de  Meechanique  Chimique,  Paris,  1878. 


LE  CHATELIER'S  PRINCIPLE.  187 

Berthelot's  theorem,  but  thereby  obtained  a  rigorous  law. 
The  modified  law  states  that  any  disturbance  of  a  system  in 
equilibrium  is  accompanied  by  forces  which  oppose  the 
disturbance.  In  particular,  if  the  temperature  of  an  equilib- 
rium system  is  raised  or  lowered,  the  resulting  modifications  are 
accompanied  by  such  heat  absorption  or  emission  as  will  reduce 
the  change  in  temperature.  For  example,  the  formation  of  a 
compound,  such  as  water,  which  has  a  positive  heat  of  forma- 
tion, is  checked  by  the  heat  liberated.  At  1124°,  99.9922%  of 
the  equivalents  of  hydrogen  and  oxygen  form  water,  while 
but  98.23%  of  water  is  formed  at  I9840.1  Such  compounds 
which  have  less  energy  than  their  constituents  are  called 
exothermic. 

If,  on  the  other  hand,  energy  is  absorbed,  as  is  the  case  when 
NO 2  is  formed  from  N2O4,  the  actual  cooling  reduces  the  amount 
formed.  Fig.  85  illustrates  how  the  amount  of  N02  in- 
creases with  rise  in  temperature.  Compounds  such  as  this, 
which  absorb  energy  in  their  formation,  are  called  endothermic 
compounds.  The  higher  the  temperature,  the  greater  the 
equilibrium  amount  of  endothermic,  and  the  less  the  amount 
of  exothermic  compounds. 

191.  Heats  of  formation  are  measured  in  a  calorimeter 
imilar  to  that  described  in  §53  unless  gases  are  among  the 
litial  or  final  products,  in  which  case  a  calorimeter  of  the 
>omb  type  (§55)  is  used.  Heats  of  solution  and  dilution 
re  usually  measured  in  a  calorimeter  of  the  first  type. 

EXPERIMENT  XXI. 
Heat  of  Solution. 

A  calorimeter  is  set  up  as  described  in  §53.  A  measured  amount 
of  water  or  other  solvent  is  poured  into  the  inner  calorimeter. 
The  substance  to  be  dissolved 2  is  carefully  weighed  and  placed 

1  Nernst,  "  Thermodynamics  and  Chemistry,"  p.  36. 

a  Any  of  the  common  salts,  acids  or  bases'  are  suitable.  Sodium  sulphate 
is  a  particularly  interesting  example  of  LeChatelier's  principle.  The  heat  of 
solution  of  the  hydrate  should  be  determined  at  about  room  temperature  and 
that  of  the  anhydrous  salt  in  water,  at  about  40°.  If  one  of  the  organic  acids 
suggested  in  Experiment  XL  is  used,  ths  results  of  the  two  experiments  can  be 
Compared. 


1  88  THERMOCHEMISTRY. 

in  a  thin-walled  test-tube.  The  latter  is  supported  in  the  solvent 
by  the  cardboard  cover.  When  the  solvent  has  been  stirred  for 
some  time  and  the  temperature  as  read  by  a  sensitive  thermo- 
meter is  approximately  constant,  readings  are  made  of  the  tem- 
perature every  minute  for  five  minutes  and  then  the  bottom  of 
the  test-tube  is  broken  by  a  glass  rod.  The  liquid  is  continually 
stirred  and  the  temperature  is  read  every  minute  until  five  minutes 
after  a  maximum,  minimum,  or  approximately  constant  temperature 
is  attained.  The  inner  calorimeter,  stirrer  and  test-tube  are  weighed 
and  their  water  equivalents,  together  with  that  of  the  thermometer, 
are  calculated  (§54).  The  temperature  observations  are  plotted 
against  the  time  and  the  radiation  correction  is  calculated  (§59). 

Let  m  =mass  of  dissolved  substance,  M  =  molecular  weight,  mw  = 
mass  of  water,  e  =  water  equivalent  of  calorimeter,  etc.,  ti  =  initial 
corrected  temperature,  t2  =  final  corrected  temperature.  The 
quantity  of  heat  emitted  is 


,-/,)  (123) 

and  the  heat  of  solution  is 


Another  method  of  determining  the  heat  of  solution  is  given  in 
Experiment  XL. 

QUESTIONS. 

1.  Why  is  it  necessary  that  the  final  solution  be   quite  dilute? 
(§§186,  187.) 

2.  Is  this  particular  solution  endothermic  or  exothermic? 

3.  Will  the  solubility  be  greater  or  less  at  a  higher  temperature? 

4.  Calculate   the    approximate    heat    developed   by   the    solution 
of  one  kilo  of  the  solute  in  100  litres  of  the  solvent. 

192.  Hess's  Law.  —  In  1840  Hess1  announced  the  law  that 
the  emission  or  absorption  of  heat  accompanying  a  chemical 
transformation  is  the  same  whether  the  transformation  from 
the  initial  to  the  final  state  proceeds  in  one  step  or  many. 
This  is  the  most  important  law  of  thermochemistry. 

A  large  number  of  the  results  given  in  Table  XXVI  were 
not  observed  directly,  but  were  calculated  by  this  law. 

The  following  examples  illustrate  the  applicability  and 
utility  of  the  law. 

193.  (A)  Heat  of  formation  of  carbon  monoxide. 

It    is   impossible   to    measure    directly   the    heat    emitted 

1  Pogg.  Ann.,  1840,  1,  9,385,  also  Ostwald's  Klassiker  No.  9. 


HESS'S    LAW.  189 

when  carbon  and  oxygen  unite  to  form  carbon  monoxide. 
Under  normal  conditions  carbon  dioxide  is  formed 

C+O2=CO2  +97600 
When  carbon  monoxide  is  oxidized  to  the  dioxide 
CO  +  O  =  CO2  +  68600 

By  Hess's  law,  the  heat  emitted  must  be  97,600  if  the  reaction 
proceeded  from  C  +O2  to  CO2  by  the  two  steps 

C+O=CO+Q;  CO  +  O=CO2  +68600 

.'.  Q,  the  heat  of  formation  of  carbon  monoxide  must  be 
29,000. 

194.  (B)  If  lime  is  treated  with  dilute  hydrochloric  acid, 
a  solution  of  chloride  of  calcium  is  obtained  and  46,000  calories 
are  liberated  per  gram  molecule 

CaO  +2HC1'  aq  =CaCl2  +H2O  +aq  +46000 

The  production  of  calcium  chloride  may  also  proceed  by  three 
steps,  the  formation  of  hydroxide  from  the  oxide  and  water 
(not  from  the  elements  as  in  Table  XXVI),  the  solution  of  the 
hydroxide,  and  finally  the  reaction  between  the  hydroxide  and 
the  hydrochloric  acid. 

CaO  +H2O=Ca(OH)2  +  21500 
Heat  of  solution  of  Ca (OH)  2  =  3 ooo 

aq  +  2HC]'aq=CaCl2+2H2O+aq  + 28000 


The  total  heat  emitted  is  46,000  calories,  or  the  same  value  that 
was  obtained  by  the  single  step. 

I95-  (Q  The  heat  of  solution  of  hydrochloric  acid  is  17,430 
calories.  The  heat  of  dilution  of  a  solution  of  the  composi- 
tion HCl+3-2H2O  is  3770  calories.  By  Hess's  law  the  heat 
emitted  when  one  gram  molecule  of  hydrochloric  acid  is 
added  to  3.2  molecules  of  water  must  equal  17430—3770  = 
13660  calories,  which  agrees  well  with  experimental  results. 


I QO  THERMOCHEMISTRY. 

(Remember  that  the  heat  of  solution  applies  to  the  formation 
of  a  very  dilute  solution.) 

196.  Thermal  Neutrality. — In  the  same  year  Hess  announced 
a  second  law.  Two  dilute  salt  solutions  may  be  mixed  with 
no  change  in  energy  (no  emission  or  absorption  of  heat). 
This  law  only  holds  for  salts  which  are  highly  dissociated,  in 
dilute  solution  (§168),  and  under  these  conditions  we  should 
expect  no  change  in  the  energy,  for  the  ions  have  an  indepen- 
dent existence  before  and  after  mixing. 

PROBLEMS  X. 

1.  Calculate    the    difference    between    the    heat    of   formation    at 
constant    pressure    and    constant    volume    of    carbon    dioxide    from 
carbon  monoxide  and  oxygen  for  (a)  one  gram  molecule  of  carbon 
dioxide;  (6)  100  grams  of  carbon  monoxide,  both  at  100°. 

2.  (a)    How  much  heat  is  developed  during  the    solution    of    50 
grams  of  sodium  hydroxide  in  10  litres  of  water?      (b)    How  much 
potassium  chloride  would  have  to  be  dissolved  simultaneously  to 
prevent  any  change  of  temperature? 

3.  The  heat  of  solution  of  hydrochloric  acid  in  2.7  molecules  of 
water  is  12,100.     Calculate  the  heat  of  dilution  for  this  concentration 
(Table  XXVI). 

4.  The  heat  of  solution  of  potassium  hydroxide  hydrate,   KOH'- 
2H2O  is  practically  zero.      What  is  the  heat  of  formation  of  this 
hydrate  from  water  and  the  solid  base?      (Find  in  Table  XXVI  the 
heat  of  solution  of  the  anhydrous  salt.) 

5.  The    formation  of    phosphoric  acid    from  yellow    phosphorus 
is  accompanied  by  an  emission  of  2386  calories  per  gram  atom  of 
phosphorus,  while  only  2113  calories  are  emitted  if  red  phosphorus 
is  used.      What  is  the  heat  of  formation  of  the  red  modification  from 
the  yellow? 

6.  State  which  of  the  following  compounds  are  (a)  endothermic; 
(b)    absolutely   stable   at   ordinary  temperatures;    (c)    stable  at   ex- 
ceedingly high    temperatures.      Ozone,    hydrogen   dioxide,    nitrogen 
peroxide,     carbon    dioxide,    methane,    carbon    bisulphide,     sodium 
carbonate. 


197.  Variations    of    Heat    Evolution    with    Temperature.— 

If  a  compound  is  formed  at  ^°  degrees,  and  raised  to  /2°, 
or,  if  the  constituents  are  heated  from  tj°  to  t°  and  then  react, 
the  final  and  initial  products  are  the  same,  and  therefore, 
by  Hess's  law,  both  procedures  give  the  same  evolution  of 
heat.  If  SI  is  the  molecular  specific  heat  of  the  constituents, 
and  52  that  of  the  compound,  and  if,  following  the  uniform 


HEAT    OF    NEUTRALIZATION.  19 1 

notation,  Ql  and  Q2  are  the  heat  energies  absorbed,  or  —  Qx 
and  —  02  are  the  evolutions  of  heat  energy  at  the  two  temperatures, 
t2  t2  t2 

-  0i  -  j  S2dt  =  -  1  S,dt  -  02,  or,  Q2  =  Q,  -  j  (5,  -  52)  dt 
If  the  specific  heats  are  constant 

If  the  equation  is  divided  by  M,  the  molecular  weight, 

q2  =  qi-  (sl  -  s2)  (t2  -  /,)     [Table  I]  (126) 

198.  Applications.   (A)  Heat  of  vaporization.     The  specific 
heat  of  water  is  approximately  unity,  that  of  saturated  water 
vapor,  at  constant  pressure,  is  .48.    The  latent  heat  of  vaporiza- 
tion (per  gram)  at  1 00°  is  53  7 .     What  is  the  latent  heatat  1 8°  ? 

537  =  qi&—  (i  —  -48)  (100—  18) 
•  '•  ^18=580 

199.  (B)  Heat  of  formation  of  one  gram  molecule  of  water 
vapor  at  1000° 

0l8  =68400  (liquid,  Table  XXVI)  -  1 8  X  580 

=  57900  (vapor) 

SH  =50=7.2;  5^0=10.1=5,   (Table  XVIII) 
•Si  =  7-2  +  3.6=  10.8 

•'•  0iooo=-  579°°-  (10.8-  10.1)982  =  -58700 
Another  method  of  solving  such  problems  is  given  in  §266. 

200.  Heat  of  Neutralization. — If  we  mix  an  acid  with  an 
ilkaline  solution,  the  concentrations  of  hydrogen  and  hydroxyl 

ions  become  greater  than  is  possible  for  equilibrium  (§258) 
and  consequently  the  majority  of  these  ions  unite  to  form 
water.  In  so  doing  they  liberate  a  certain  amount  of  heat 
energy.  The  amount  of  heat  liberated  per  gram  molecule 
of  water  formed  is  called  the  heat  of  neutralization.  Evi- 
dently this  quantity  should  be  the  same  for  all  completely  dis- 
sociated acids  and  bases,  and  all  the  common  strong  acids  and 
alkalies  give  about  13,800  calories.  If  either  base  or  acid  is 


192 


THERMOCHEMISTRY. 


feebly  ionized,  the  heat  of  neutralization  may  be  greater  or 
less  than  this  value  according  as  heat  is  emitted  or  absorbed, 
in  the  progressive  ionization  which  must  accompany  the 
neutralization. 

20 1.  Heat  of  Ionization. — Evidently  the  heat  of  neutraliza- 
tion, 13,800  calories,  is  the  energy  required  to  dissociate  one 
gram  molecule  of  water  into  its  ions  (for  the  dissociation  to  be 
permanent,  this  gram  molecule  of  water  must  be  mixed  with 
a  very  great  volume  of  undissociated  water  (§§258,  328). 
Table  XXVI  shows  that  68,400  calories  are  required  to  form 
one  gram  molecule  of  liquid  water  from  its  elements.  There- 
fore, 68400  —  13800  =  54600  calories  is  the  heat  of  formation 
of  one  gram  molecule  of  hydrogen  and  hydroxyl  ions. '  It  is 
known  that  very  little  energy  is  required  to  ionize  gaseous 
hydrogen .  dissolved  in  water.1  Therefore,  54,600  calories  is 
approximately  the  heat  of  formation  of  one  equivalent  of  OH' 
ions  (17  grams).  (For  meaning  of  symbols,  see  §287  ) 

TABLE   XXVIII. 
Heats  of  Formation  of  Ions." 

(The  unit  is  the  large  calorie  =  1000  common  calories.) 


Ion 

Symbol 

Ion 

Symbol 

Hydrogen  

TT  . 

o 

Copper  .  . 

Cu" 

-i  <  8 

Potassium  
Sodium   
Lithium  

K- 

Na- 
Li- 

61.9 

57-5 

62.0 

Copper  
Mercury  
Silver  

Cu- 

?g' 
Ag' 

-16. 
-19.8 

—2  C.  7 

Ammonium  .... 

NH4' 

32.  8 

Lead  

Pb" 

.  c 

Hydroxylamine 

NH4O' 

-}  1      C 

Tin 

Sn" 

3? 

Magnesium    . 

Mg" 

TOO 

Chlorine 

Cl' 

•2  Q     -7 

Calcium    

Ca" 

IOQ 

Bromine   

Br' 

28   2 

Aluminum  
Manganese  ..... 
Ferrous  ion  

A1-" 
Mn- 
Fe" 

121 
50.2 
22.2 

lod'ne    
Sulphate  
Sulphite  

V 

SO/' 
SO3" 

I3-1 

214.4 

I  ^  I  •  ^ 

Ferric  ion 

Fe"- 

—  o  7 

Nitrous 

NO2' 

2  7 

Cobalt 

Co" 

I  7 

Nitric 

NO/ 

Nickel    
Zinc    

Ni" 
Zn" 

16.- 

•?  c.  i 

Carbonate    
Hydroxyl       .... 

CO/ 
HO' 

161.1 

?A     6 

Cadmium  

Cd" 

18.4 

*  Jiiptner,  i,  p.  i 
2  Jiiptner,  i,  p.  i 

79- 
So. 

HEAT    OF    NEUTRALIZATION.  193 

Other  ionic  heats  of  formation  may  now  be  calculated. 
For  example,  KOH'aq  =  116500  calories  (Table  XXVI). 
The  ionic  heat  of  formation  of  K'  ions  must  be  116500  —  54600 
=  61900.  KCl'aq  =  104300  —3100  =  101200.  Hence  the  ionic 
heat  of  formation  of  chlorine  ions,  (Cl')  is  101200—61900  = 
39300  calories.  In  this  manner  Table  XXVIII  has  been 
constructed. 

EXPERIMENT  XXII. 

Heat  of  Neutralization. — A  calorimeter  is  prepared  as  described  in 
§53.  Equivalent  solutions  (for  example,  twice-normal)  of  the 
assigned  acid  and  base  are  also  prepared.  One  solution  is  placed  in 
the  inner  glass  calorimeter  vessel  and  the  other  is  poured  into  a  large 
test-tube  which  is  held  by  the  cardboard  cover  about  a  centimeter 
from  the  bottom  of  the  inner  calorimeter.  A  delicate  thermometer 
and  a  glass  stirrer  are  also  placed  in  the  inner  calorimeter  vessel. 
The  outer  solution  is  constantly  stirred,  and  when  sufficient  time  has 
elapsed  for  the  temperature  to  become  constant,  the  temperature 
is  read  to  tenths  of  the  smallest  division  (e.g.  .01°),  every  minute  for 
five  minutes,  and  then  the  bottom  of  the  test-tube  is  broken  with 
a  glass  rod.  The  liquid  is  stirred  gently  but  steadily  throughout 
the  experiment  and  the  temperature  is  read  every  minute  until 
five  minutes  after  the  maximum  temperature  is  attained. 

The  temperatures  are  plotted  against  the  times  and  the  radiation 
correction  is  calculated  as  described  in  §59.  The  heat  developed  is 
calculated  from  the  corrected  change  in  temperature  and  the  water 
equivalent  of  the  calorimeter  (§54)  and  contents.  The  specific 
heats  of  the  solutions  may  be  taken  as  the  same  as  that  of  the  con- 
tained water,  with  little  error.  The  ratio  of  the  heat  developed  to 
the  number  of  gram  equivalents  of  salt  formed  is  the  molecular  heat 
of  neutralization. 


QUESTIONS. 

1.  Why  should  the  result  be  independent  of  the  kind  of  acid  or 
base  employed? 

2.  Calculate  the  rise  in  temperature  upon  mixing  (a)  a  litre  of  the 
assigned  base  and  a  litre  of  the  assigned  acid;  (6)  a  litre  of  a  10% 
solution  of  the  base  with  a  litre  of  a  10%  solution  of  the  acid;  (c)  a 
litre  of  one-half  normal  solution  of  the  base  with  a  litre  of  half-normal 
solution  of  the  acid. 

3.  Write   the   ordinary   chemical   equation   for  the   reaction   and 
also  the  ionic  equation. 

202.  Heat  of  Combustion. — The  heat  emitted  during  the 
complete  oxidization  of  a  unit  mass  of  substance  is  called 
the  heat  of  combustion.  The  unit  of  mass  in  physical  chemistry 


194 


THERMOCHEMISTRY. 


TABLE  XXIX. 

Heat  of  Formation  and  Heat  of  Combustion  of  Organic  Compounds.' 

(The  unit  is  the  large  calorie  =  1000  common  calories). 


Compound 

Symbol 

Heat  of  Formation 

Heat  of 
Combus- 
tion 

(Vapor) 

(Liquid)  \  Solution 

Methane 

CH4 
C2H6 
C3H8 
C2H4 
C2H2 
CH3OH 
C2HSOH 
CSHI0OH 
C3H803 
C2H402 
Cl8H3602 
CeHe 
CIOH8 
C6H60 
C7H602 
C7H603 
C6H7N 
C6H,N02 
CIOHI60 

18.9 
23-3 
30-5 
—14.6 
-58.1 
53-3 
59-8 
80.9 

112.  1 

24of? 

—11.3 

—  22.  8f 

36.8f 
94-  2f 
132.1! 

"8o'.3t' 

213-5 
372-3 
528.4 

34i.i 
3IS-7 
170.6 

325-7 
793-9 
397-2 
209.4 
2712. 
784 
1242. 
736 
773 
735 
818 

733 
1414 

!355 

Ethane  
Propane  
Ethylene  
Acetylene  
Methyl  alcohol  . 
Ethyl  alcohol  .  . 
Amyl  alcohol  .  . 
Glycerine  
Acetic  acid  
Stearic  acid  .... 
Benzol  
Naphthalene  .  .  . 
Phenol  
Benzoic  acid  .  .  . 
Salicylic  acid  .  .  . 
Aniline  
Nitrobenzene.  .  . 
Camphor 

61.7 
69.9 
91.6 
161.7 
117.2 

63-7 
72.4 

94-4 
167.1 
117.6 

—27.4 

34-5 
91.9 

•^  • 
87.7 

125-7 

1  1.2- 

5-i 

Cane-su^ar 

is  usually  the  gram  molecule.  In  engineering  it  is  usually 
one  gram  or  one  pound,  and  the  heat  of  combustion  is  often 
called  the  heat  value  or  calorific  power.  Since  most  compounds 
are  exothermic,  the  heat  of  combustion  is  usually  less  than 
that  of  the  oxidation  of  the  separate  constituents. 

There  are  two  rules  which  are  often  used  for  the  approximate 
calculation  of  the  heat  of  combustion  of  organic  substances. 
The  first  rule  (Welter's)  states  that  the  heat  of  combustion  per 
gram  molecule  is  equal  to  the  heat  of  formation  of  the  oxides 
of  the  elements  which  remain  after  all  the  oxygen  and  the 
equivalent  amount  of  hydrogen  for  the  formation  of  water 
have  been  subtracted.  The  second  rule  requires  the  subtrac- 
tion of  the  oxygen  and  the  equivalent  amount  of  carbon  for 

1  Reychler-Kiihn,  pp.  177-181. 
t  S  lid. 


HEAT    OF    COMBUSTION.  1 95 

the  formation  of  carbon  dioxide  and  then  the  heat  of  oxida- 
tion of  the  remainder  is  calculated. 

Illustration. — Heat  of  combustion  of  stearic  acid  (C^H^O,,). 
By  the  first  rule,  we  subtract  H4O2,  leaving  C^H^.  The 
heat  of  combustion  of  this  carbon  and  hydrogen  is  i6X 
97,600  +  16X68,400  =  2,600,000.  By  the  second  rule  we 
would  subtract  CO2  and  the  heat  of  combustion  of  the  remain- 
der is  15  X 97, 600 +  18X68, 400  =  2,700,000.  The  experimental 
result  is  2,700,000. 

203.  The  heat  of  combustion  will  depend  upon  the  physical 
conditions  under  which  the  combustion  takes  place.  If  the 
water  vapor  formed  is  allowed  to  condense,  the  heat  of  con- 
densation and  that  of  cooling  of  the  resulting  water  will  be 
added  to  the  true  heat  of  combustion.  If  the  combustion 
takes  place  at  constant  pressure,  the  heat  of  combustion  per 
gram  molecule  will  be  greater  than  that  at  constant  volume 
by  wR6  =  2w(2 73  +/)  (§184),  where  w  is  the  decrease  in  the 
number  of  gram  molecules.  For  example,  in  the  combustion 
of  hydrogen 

H2+O=H2O 

w  is  1/2  if  the  water  remains  as  vapor  and  i  1/2  if  it  is  liquid. 
Approximate  values  of  the  heats  of  formation  may  often  be 
calculated  from  heats  of  combustion.  Methane,  CH4,  for 
example,  has  a  heat  of  combustion  of  213,500  calories  (Table 
XXIX).  If  the  constituent  hydrogen  and  oxygen  were 
oxidized,  the  heat  emitted  would  be 

C  +O2  =CO2  +97, 600  ;H4  +  2O2  =  2H2O  +  2X68400 
or  a  total  of  234,400  calories.  Since,  by  Hess's  law,  the  total 
amount  of  heat  must  be  the  same  if  the  intermediate  product, 
methane,  is  formed,  the  heat  of  formation  of  methane  must  be 
234400  —  213500=20900  calories,  which  is  approximately  the 
value  found  by  direct  experiment. 

PROBLEMS  XI. 

i.  If  a  piece  of  iron  is  placed  in  a  solution  of  copper  sulphate, 
copper  is  deposited  and  an  equivalent  amount  of  iron  goes  into 
solution.  Calculate  from  Table  XXVI  the  heat  emitted  per  gram 
of  copper  deposited. 


196  THERMOCHEMISTRY. 

2.  The  specific  heat  of  liquid  benzol  is  about  .38  and  that  of  the 
vapor  is  about  .29.      What  is  the  approximate  latent  heat  of  vapor- 
ization at  20°,  if  the  latent  heat  at  80°  is  93  ? 

3.  Assuming  12,000  as  the  heat  of  formation  of  ammonia  (at  con- 
stant pressure)  at  100°,  calculate  the  heat  of  formation  at  1000°. 

4.  Calculate  the  heat  of  formation  of  a  dilute  aqueous  solution 
of  (a)   sodium  chloride,   (6)   sodium  sulphide,  from  the  ionic  heats 
(Table  XXVIII)  and  also  from  the  combined  heats  of  formation  and 
solution  (Table  XXVI). 

5.  450  grams  of  a  solution  which  contained  9.12  grams  of  hydro- 
chloric acid,  the  temperature  of  which  was  18.22°,  was  mixed  with 
450  grams  of  a  solution  which  contained   10  grams  of  sodium  hy- 
droxide in  a  calorimeter  whose  water  equivalent  was  13.      The  initial 
temperature  of  the  latter  solution  was   18.61.     The  final  corrected 
temperature    of    the    mixture    was    22.17.     Calculate    the    heat    of 
neutralization  per  gram  molecule.      (Thomsen.) 

6.  Calculate  the  approximate  heat  of  formation  of  (a)  ethane,  (6) 
acetylene,  from  their  heats  of  combustion. 

7.  Calculate  the  heat  of  combustion  of  (a)  ethane,  (b)  benzoic  acid, 
by  both  rules  of  §202,  and  compare  the  results  with  the  experimental 
values  given  in  Table  XXIX. 

EXPERIMENT  XXIII. 
Heat  of  Combustion  of  a  Solid  Substance.     Use  of  Bomb  Calorimeter. 

Full  directions  for  both  the  constant  volume  and  constant  pressure 
combustion  calorimeters  are  given  in  §§55  and  56. 

If  a  substance  of  indefinite  composition,  such  as  bituminous  coal, 
is  used  in  this  experiment,  a  definite  chemical  compound,  such  as 
ethyl  alcohol  or  benzol,  should  be  used  in  the  next  experiment. 

Express  your  results  in  (a)  calories  per  gram,  (6)  B.  T.  U.  per 
pound  (§7),  and  (c),  if  the  composition  is  definite,  in  calories  per  gram 
molecule.  With  (c),  calculate  the  heat  of  combustion  from  the 
composition  by  both  rules  of  §202  and  state  the  difference  in  value 
for  constant  pressure  and  constant  volume. 

QUESTIONS. 

1.  Calculate  the  heat  value  of  (a)  one  kilo  of  this  substance,  (6) 
one  short  ton  in  B.  T.  U.  per  lb.,  (c)  the  mechanical  energy  equivalent 
to  the  latter. 

2.  What  error  would  be  caused  by  (a)  an  error  of  20  in  the  water 
equivalent  ?      (6)  allowing  a  current  of  5  amperes  to  flow  through  a 
platinum  wire  of  2  ohms  resistance  for  5  seconds?      (c)   Neglecting 
the  radiation  correction  ? 

EXPERIMENT  XXIV. 
Heat  of  Combustion  of  a  Liquid  or  Vapor.     Use  of  Junker  Calorimeter. 

Full  directions  for  the  use  of  the  Junker  calorimeter  for  both 
liquid  and  gaseous  substances  are  given  in  §§57  and  58.  It  is 
particularly  interesting  to  use  a  liquid  of  known'composition,  such 
as  ethyl  alcohol,  benzol,  etc.,  or  a  gas  of  definite  composition. 


HEAT    OF    COMBUSTION. 


I97 


Express  your  results  in  (a)  calories  per  litre  (if  a  gas,  the  volume 
must  be  under  standard  conditions  (§4),  (6)  B.  T.  U.  per  gallon  or 
cubic  foot,  and  (c),  if  the  composition  is  definite,  in  calories  per 
gram  molecule.  With  (c)  calculate  the  heat  of  combustion  by  both 
rules  of  §202  and  state  the  difference  in  value  for  constant  pressure 
and  constant  volume. 


QUESTIONS. 

1.  Is  the  heat  value  of  a  gas,  per  gram,  definite? 

2.  What  difference  would  there  be  in  the  result 
vapor  escaped  without  condensing? 

3.  Why  is  no  radiation  correction  necessary? 
4- 


Per  litre? 
if  all  the  water 


What  is  the  heat  value  of  100  litres  of  this  substance? 


CHAPTER  V. 
LIGHT.1 

We  shall  confine  this  chapter  to  a  study  of  certain  optical 
phenomena  which  are  of  particular  interest  and  importance  to 
chemists. 

204.  Emission  of  Light. — The  electro-magnetic  waves  in  the 
ether  which  produce  in   the  eye  the  sensation  of   light  are 
emitted  by  incandescent  solids  or  liquids,  by  gases  subjected 
to  electric  forces,  and  by  bodies  undergoing  certain  chemical 
changes.     Incandescent,  solid,  or  molten  iron,  the  mercury 
vapor  lamp,  and  oxidizing  phosphorus  may  be  cited  as  ex- 
amples.    Secondary   light    waves    are    often    emitted    when 
bodies  are  subjected  to  light.     The  fluorescence  of  a  solution 
of  sulphate  of  quinine  is  an  illustration.     The  emission  of 
ether  waves  by  physical  or  chemical  causes  other  than  high 
temperature   is  called  luminescence.     We   shall   consider  the 
various  types  of  luminescence  after  we  have  considered  pure 
thermal  radiation. 

205.  Thermal  Radiation. — We  shall  define  a  black  body  as 
one  which  completely  absorbs  all  the  ether  waves  which  fall 
upon  it.     The  absorbing  power  of   any  body  is  the   fraction 
which  it  absorbs,  of  the  energy  of  the  ether  waves  falling  upon 
it,  and  the  emissive  power  or  emissivity  is  the  ratio  of  the  amount 
of  energy  radiated  by  the  body  to  the  energy  which  would  be 
radiated  by  a  similar  black  body.    In  1 859  KirchofI  formulated 
the  following  law.    "  At  a  given  temperature,  the  ratio  between 
the  emissive  and  absorbing  powers,  for  a  given  wave  length, 
is  the  same  for  all  bodies."2     Ritchie3  had  previously  found 

1  General  references  for  Chapter  V:    Wood,   "Physical    Optics";    Drude, 
"Theory  of  Optics"; Preston,  "Theory  of  Light";  Mascart  "Optique";  Winkel- 
mann,   1906,  vol.  vi;   Rapports  Paris  Congres,  1900,  vol.  ii;  Kayser,  "Spectro- 
scopie";  Landauer,  "Spectrum  Analysis." 

2  Pogg.  Ann.,  1860,  cix,  p.  275. 

3  Pogg.  Ann.,  1833,  xxviii,  p.  378. 

198 


THERMAL    RADIATION.  1 99 

experimentally  that  the  greater  the  absorption  of  a  body,  the 
greater  also  was  the  emission.  A  black  body  therefore  gives 
also  greater  emission  than  any  other  body. 

206.  Lamp-black  and  platinum-black  are  approximate  black 
bodies.     A  small  aperture  in  a  hollow  body,  at  a  constant 
temperature,  is  a  very  perfect  black  body.     Whatever  the 
nature   of  the  walls,   any  waves  which   enter  the  aperture 
are   either   absorbed,  or   suffer  multiple  reflections,  without 
emerging  to  any  appreciable  extent.     Radiations  from  within 
find   even   greater   difficulty   in   escaping.     Wood   gives   the 
following  illustration.     A  fragment  of  decorated  china  shows 
the  figure  very  clearly  when  heated  to  incandescence,  because 
of  the  greater  absorption  and  consequent  greater  emission  of 
the  pigment.     If,  however,  the  pigment  is  heated  to  the  same 
temperature  in  the  artificial  black  body  just  described  and 
is  examined  through  the  aperture,  the  figure  is  not  distinguish- 
able because  the  inferiority  of  emission  of  the  plain  china  is 
compensated  by  greater  reflection  of  the  radiations  from  the 
interior  walls. 

207.  The    Stefan-Boltzman    Law   for    the    Total    Thermal 
Radiation. — Accompanying  the  ether  waves  which  affect  the 
eye  are  usually  longer  waves  which  can  only  be  detected  by 
their  thermal  effects  and  shorter  waves  which  may  be  studied 
by    their    action    on    a    photographic    plate.     Stefan,1    from 
observations,  and  Boltzman,2  from  theoretical  considerations, 
deduced  the  following  law  connecting  the  absolute  temperature 
of  a  body  and  the  total  energy  of  the  ether  waves  which  it 
emits  because  of  its  high  temperature.     If  W  is  the  energy 
emitted  by  an  area  A,  at  absolute  temperature  6,  in  T  seconds, 

W=oAT6*  (127) 

where  a  is  a  constant.  Kurlbaum^  verified  the  law  ex- 
perimentally and  found  that 

<T=  1.71  Xio-s-6^  =  .171  Xio-'2  watts  =  . 408  X  io-'2-aloneS 
sec.  sec. 

*  Ber.  d.  K.  Akad.,  Wien,  1879,  Ixxix.  B.,  p.  391. 

2  Wied.  Ann.,  1884,  xxii,  p.  291. 

3  Wied.  Ann.,  1898,  Ixv,  p.  746.     Also  see  Fery,  C.  R.,  1909,  cxlviii,  p.  915. 


200 


LIGHT. 


Approximate  Derivation. 


208.  Maxwell  proved  theoretically,  and  Lebedew,  and  Nichols 
and  Hull,2  demonstrated  experimentally,  that  ether  waves  which 
fall  on  a  black  body  exert  upon  unit  area  of  that  body,  a  pressure 
equal  to  their  energy  per  unit  volume.  A  radiating  black  body 
must  experience  a  similar  reaction.  If  we  designate  by  B  the  energy 
per  second  of  the  ether  waves  falling  upon  a  body,  inclosed  in  a  black 
body  at  constant  temperature,  6,  the  pressure  normal  to  the  surface 
is  1/3  B,  for  the  radiation  is  diffuse;  that  is,  from  all  directions;  and 
we  saw  that  the  pressure  due  to  an  analogous,  diffuse,  molecular 

bombardment  was  equivalent  to  that 

.,£  due  to  one-third  of  all  the  molecules 

moving  perpendicular  to  the  surface 
(§98).  This  normal  pressure  might 
theoretically  perform  an  equivalent 
amount  of  mechanical  work.  If,  in 
Equation  101,  we  substitute  B  for  the 
gain  of  internal  energy,  —I,  and  i  /3  B 
for  W  we  have 


100 


dB 
B 


'  B2      024 

or  the  energy  per  second  is  propor- 
tional to  the  fourth  power  of  the 
absolute  temperature. 

209.  Wien's  Displacement  Law. 
— Fig.  7 2^  represents  the  observa- 
tions of  Lummer  and  Pringsheim. 
The  ordinates  are  proportional  to 
the  energy  and  the  abscissas  to  the 
wave  lengths  (in  /*  =  .ooi  mm.).  The  short  heavy  line  repre- 
sents the  range  of  the  visible  spectrum.  The  number  against 
each  curve  is  the  absolute  temperature  of  the  radiating  body. 
Notice  that  not  only  does  the  area  beneath  each  curve — that  is, 
the  total  energy — increase  very  rapidly,  when  the  temperature 

1  Haber,  "Thermodynamics  of  Gas  Reactions,"  p.  282. 

2  Phys.  Rev.,  1901,  xiii,  p.,.293. 

3  Lummer,  Rapports,  Paril  Congres,  1900,  ii,  p.  83. 


1       a      3      4 
FIG.  72. 


WIEN'S  SECOND  EQUATION. 


201 


is  raised  (proportional  to  the  fourth  power  as  we  have  just 
i),  but  the  wave  length,  Am,  at  which  the  energy  is  a  maxi- 
mum, decreases.  Wien1  showed  from  theoretical  considera- 
tions that 

XmO  =  constant  =  c  (128) 

or,  the  wave  length  for  which  the  energy  is  a  maximum  is 
inversely  proportional  to  the  absolute  temperature.  This  is 
known  as  the  displacement  law  and  it  has  been  remarkably 
verified  by  the  experiments  of  Lummer  and  Pringsheim2  and 
>thers.  The  mean  value  of  the  above  constant,  for  a  black 
body,  is  2940  (Xm  in  /*) . 

210.  Planck's  LaW  for  the  Spectral  Distribution  of  Energy. — Planck3 
has  derived  the  following  equation  for  B,  the  radiant  energy,  per 
second,  per  square  centimeter;  for  wave  lengths  between  A  and  A  +dA, 
rhere  d\  is  a  very  small  difference  of  wave  length,  at  the  absolute 
temperature  6. 


(129) 


/•hen  c-i  and  c2  are  constants  and  e  is  the  base  of  natural  logarithms. 
211.  Wien's  Second  Equation. — If  the  wave  lengths  consid- 
ered are  very  short,  as  is  the  case  in  optical  pyrometry,  the 
parenthesis  of  Equation  129  may  be  considered  unity.     This 
simpler  but  less  accurate  equation  was  discovered  by  Wien,4 
>efore  Planck's  work.     Taking  logarithms  of  both   sides   of 
ds  simpler  equation  (§§13,  14),  we  have 


InB 


I  —  5  ln>*  —  - 


If  we  measure  the  radiation  of  a  definite  wave  length,  at  two 
temperatures,   Ol  and  02,  and  subtract  the  two  logarithmic 
[uations, 


B2 


_ 

e 


(130) 


1  Ber.  d.  K.  Akad.,  Berlin,  1893,  p.  55;  Wood, 

2  Verb.  Deutsch.  Phys.  Gesel.,  1899,  i,  p.  218. 

3  Verb.  Deutsch.  Phys.  Gesel.,  1900,  ii,  p.  202. 

4  Wied   Ann..  1896,  Iviii,  p.  662. 


Phys.  Optics,"  p.  470. 


202  LIGHT. 

If  Equation  129  is  solved  for  the  maximum  value  of  B,  c2 
will  be  found  equal  to  5  c  (Equation  128) ;  hence,  cz  =  14,700. 

212.  Optical   Pyrometers.1 — Optical  pyrometers  are  instru- 
ments for  determining  the  temperature  of  distant  bodies  from 
the  radiations  which  they  emit.     They  may  be  divided  into 
two  types.     In  the  first  type  (Fery  thermo-electric  telescope) 
the  total  radiation  is  concentrated  upon  a  thermocouple  by 
a  fluorite  lens  and  the  total  energy,  B,  is  measured  by  the 
deflection  of  a  galvanometer  connected  to  the  thermocouple. 
The  ratio  of  the  temperatures  of  two  bodies  is,  by  Equation 
127,  proportional  to  the  fourth  root  of  the  energy. 

213.  The  second  type  of  optical  pyrometer  depends  upon 
matching  the  intensity  of  the  light  from  the  hot  body,  of  a 
particular  color,  with  the  similar  light  from  a  lamp  in  the 
instrument.     These   instruments    are    therefore    based    upon 
Equation  130,  although  all  are  calibrated  empirically  from  a 
black  body  at  known  temperatures.     Red  glasses  are  usually 
used  which  limit  the  radiation  to  about  .65^. 

In  the  Le  Chatelier  pyrometer  the  light  from  the  hot  body 
is  reduced  by  a  calibrated  iris  diaphragm,  to  equality  with 
the  light  from  a  small  gasoline  lamp. 

The  Fery  absorption  pyrometer  substitutes  absorption 
wedges  for  the  variable  diaphragm. 

The  Wanner  pyrometer  uses  a  small  incandescent  lamp  and 
varies  the  amount  of  light  from  the  hot  body  by  means  of  two 
Nichol's  prisms. 

214.  The  most  accurate  optical  pyrometer  is  that  devised  by 
Holborn    and    Kurlbaum.2     The    current    through     a     small 
incandescent  lamp  is  varied  until  its  light  is  equal  to  that  from 
the  hot  body.     When  such  equality  has  been  attained,  the 
filament  and  the  hot  body  must  be  at  the  same  temperature 
(if  the  radiations  of  both  are  similar  to  the  radiation  from  a 
black  body) .     If  necessary,  one  or  more  red  glasses  are  inter- 

1  Excellent   discussions  of  optical   pyrometers  are   given   by   Waidner  and 
Burgess  in  Bui.  Bureau  of  Stand.,  1905,  i,  p.  189;  and  by  Haber,  "Thermo- 
dynamics," pp.  281-291. 

2  Ber.  Akad.  d.  Wiss.,  Berlin,  1901,  p.  712.     (The  sale  of  this  instrument 
in  the  United  States  is  prevented  by  patent  litigation.) 


UNIVER 


OPTICAL    PYROMETERS. 


203 


posed  to  protect  the  eye.  The  instrument  is  calibrated  by 
determining  the  current  necessary  to  bring  the  filament  to 
different  temperatures.  The  current  through  the  filament  is 
adjusted  until  the  filament  has  the  same  appearance  as  a 
standard  black  body,  and  the  temperature  or  the  black  body 
is  determined  by  an  air  thermometer,  or  thermocouple.  A 
system  of  lenses  assists  the  comparison.  Having  such  a 
calibration,  the  instrument  may  be  used  to  determine  the 
temperature  of  any  body  whose  radiation  is  similar  to  that  of 
a  black  body.  The  error  in  determining  the  temperature  of 
so  extreme  a  departure  from  a  black  body  as  brightly  polished 
platinum,  amounts  to  about  74°  at  950°,  but  for  most  bodies  it 
is  very  much  less. I 

EXPERIMENT  XXV. 

Study  of  Various  Methods  of  High-temperature  Measurements. 
Electric  Furnace,  Thermocouple,  Platinum  Resistance  Thermometer, 
Optical  Pyrometer. 

A  hollow  black  body  is  heated  electrically  and  the  temperature 
of  the  interior  is  determined  by  means  of  a  calibrated  thermocouple. 
A  platinum  resistance  thermometer  is  calibrated,  and  also  an  in- 
candescent lamp  is  calibrated  for  use  as  an  optical  pyrometer. 

Electric  Furnace. — The  electric  furnace  (see  Fig.  73)  consists  of  a 
thin  porcelain^  cylinder  about  15  cm.  long  and  10  cm.  in  diameter 
upon  which  is  wound  about  5  m.  of  No.  22  "Nichrome"3  wire,  if  a 
2 20- volt  supply  is  to  be  used.  Whatever  the  voltage,  the  winding 
must  be  such  as  to  consume  about  half  a  kilowatt.  The  ends  of  the 
cylinder  are  closed  by  porcelain  caps  with  proper  apertures  and  the 
whole  is  surrounded  by  many  layers  of  asbestos. 

Heating  and  cooling  must  be  very  gradual  so  that  the  thermo- 
couple and  platinum  thermometer  may  acquire  the  temperature  of 
the  furnace.  The  highest  temperature  should  not  exceed  1000°. 

Thermocouple. — A  platinum, — platinum,  10%  rhodium,  thermo- 
couple 4  should  be  connected  to  a  galvanometer  through  a  key  and  such 
a  resistance  as  will  keep  the  deflection  on  the  scale  at  the  highest 
temperature.  The  galvanometer  with  resistance  should  be  calibrated 
as  a  voltmeter  (§49).  (The  potentiometer  method  described  in  §73 
is  more  accurate,  but  does  not  permit  as  rapid  observations.)  The 
chart  or  table  accompanying  the  couple  gives  the  temperature  of  the 

1  See  end  of  articles  by  Waidner  and  Burgess,  and  Haber. 

2  The  author  will  be  pleased  to  give  any  additional  information  desired  in 
regard  to  porcelain,  etc. 

3  Driver-Harris  Co.,  Newark,  N.  J. 

*  Excellent,  calibrated,  Hereaus  thermocouples  and  Hereaus  platinum  wire 
may  be  obtained  from  Charles  Engelhardt,  Hudson  Terminal  Building,  New 
York  City. 


2O4 


LIGHT. 


PLATINUM    RESISTANCE    THERMOMETER.  205 

hot  junction  when  the  electromotive  force  is  known.  (The  cool 
junction  should  be  in  ice  and  water.) 

Platinum  Resistance  Thermometer. — The  platinum  resistance 
thermometer  consists  of  a  coil  of  fine  platinum  wire  (for  example,  50 
cm.  of  No.  30)  wound  on  a  porcelain  frame  and  surrounded  by  a  glazed 
porcelain  tube.  The  coil  constitutes  one  arm  of  a  Wheatstone's  bridge 
(§§66-68).  A  pair  of  dummy  leads  are  connected  to  an  adjoining 
arm  (see  figure)  and  compensate  for  the  heating  of  the  lead  wires. 
A  suitable  switch  connects  the  galvanometer  to  either  the  bridge 
or  the  thermocouple. 

Optical  Pyrometer. — The  optical  pyrometer  consists  of  a  lens 
(object-glass)  which  focuses  upon  the  tip  of  the  filament  of  a  mina- 
ture  incandescent  lamp,  the  interior  of  the  nearly  inclosed  furnace 
(ideal  black  body).  The  current  through  the  filament  is  adjusted 
until  the  tip  of  the  filament  is  invisible  against  the  image  of  the 
furnace.  When  this  is  true,  both  must  be  emitting  similar  light, 
and  therefore  (§205)  they  must  be  at  the  same  temperature.  A 
small  eye-piece  aids  in  observing  the  tip  of  the  filament.  Calibrate 
the  incandescent  lamp  filament,  i.e.,  find  the  current  necessary  to 
heat  the  tip  of  the  filament  to  different  temperatures.  The  tem- 
perature of  the  filament  is  determined  by  finding  the  temperature  of 
the  furnace,  by  the  thermocouple,  when  the  two  have  the  same 
temperature. 

Observations. — While  the  furnace  is  slowly  heating,  and  also  while 
it  is  slowly  cooling,  observe  at  .frequent  intervals,  (^4)  the  galvan- 
ometer deflection  with  the  thermocouple,  and  the  resistance  in  the 
galvanometer  circuit;  (B)  the  resistance  of  the  platinum  ther- 
mometer; (C)  the  current  through  the  filament.  The  three  obser- 
vations should  be  made  in  succession  and  the  times  of  each  recorded. 
(D)  Calibrate  the  galvanometer  as  described  in  §49.  (E)  If  time 
permits,  use  the  optical  thermometer  to  determine  the  temperature 
of  various  distant,  brightly-heated  bodies,  e.g.,  melted  silver,  iron, 
or  copper.  An  image  of  the  hot  body  is  formed  upon  the  tip  of  the 
filament,  and  the  current  through  the  latter  is  adjusted  until  the  two 
are  indistinguishable.  The  temperature  corresponding  to  this 
current  is  obtained  from  the  calibration  (see  /  below) . 

Report. — (a)  Tabulate  readings.  (6)  Plot  the  three  sets  of 
readings  against  the  time,  (c)  From  (B)  make  a  plot  giving  the 
resistance  of  the  platinum  thermometer  plotted  against  the  time. 
Determine  R0  and  the  two  constants,  a  and  6,  of  the  equation, 

>)  (131) 


by  choosing  three  values  of  t,  and  the  corresponding  values  of  Rt, 
substituting,  and  solving  the  three  simultaneous  equations,  (d)  Make 
a  third  plot  with  resistances  as  abscissas  and,  for  ordinates,  the  plat- 
inum temperatures,  as  given  by  Callender's  equation, 

pt  =  ioo     *  _j°  (132) 

where  R0  is  the  exterpolated  resistance  at  o°  and  Ri  is  the  resistance 
at  100°.  Transfer  to  this  plot,  also,  the  readings  of  the  true  tem- 
perature, t,  and  determine  the  mean  value  of  Callender's  difference 
constant,  5,  by  applying  at  several  points  the  equation 

'-  ('33) 


206  LIGHT. 

For  pure  platinum,  5  is  1.50.  The  platinum  resistance  thermome- 
ter is  the  most  accurate,  convenient,  method  of  measuring  tem- 
perature below  1000°. 

(e)  Construct  a  curve  which  gives  the  temperature  of  the  tip  of 
the  incandescent  lamp  filament  plotted  against  the  current.  (/) 
Finally,  determine  by  the  latter  curve  the  temperatures  of  any 
bodies  tested  with  the  optical  pyrometer,  and  record  the  results. 

QUESTIONS. 

1.  Would   you   expect   the   platinum   resistance   thermometer   to 
attain  a  slightly  higher,  or  a  slightly  lower  temperature  than  the 
thermocouple?     Explain. 

2.  Is  any   correction   required   for  the   absorption  of  the  lenses 
in  such  an  optical  pyrometer?     Explain. 

3.  What    small   errors   were    disregarded   in   calibrating   the   gal- 
vanometer? 

215.  Spectrum  of  Gases. — We  have  considered  thus  far  the 
radiation  from  solids  and  liquids.     It  is  doubtful  if  gases  or 
vapors  give  off  appreciable  light  at  any  attainable  tempera- 
ture, unless  they  are  simultaneously  subjected  to  chemical 
changes  or  electrical  forces  (luminiscence)  .x 

When  a  gas  is  subjected  to  proper  electric  forces,  it  emits 
light  of  certain  definite  wave  lengths  which  are  characteristic 
of  the  particular  gas.  If  a  light  from  a  gas  is  analyzed 
by  a  spectrometer  (§61)  narrow,  isolated,  bright  images  of 
the  slit  ("lines")  are  seen  in  different  parts  of  what  would 
be  a  continuous  spectrum  if  the  gas  was  replaced  by  an  incan- 
descent solid.  The  most  convenient  method  of  producing 
electro-luminiscence  is  to  inclose  the  gas  in  a  tube  provided 
with  platinum  electrodes.  If  the  tube  is  exhausted  to  a  partial 
vacuum,  the  gas  will  transmit  a  much  greater  current,  at  a 
lower  potential,  than  it  will  at  atmospheric  pressure. 

216.  Spectrum  of  Vapors. — The  spectrum  of  vapors  is  often 
obtained  in  a  similar  manner,  but  there  are  also  other  methods 
which  are  available  in  many  cases.     The  spectrum  of  a  metallic 
vapor  can  be  obtained  by  making  a  rod  or  block  of  the  metal 
one  pole  of  an  electric  arc.     The  metal  is  vaporized  and  the 
vapor  emits  the  characteristic  spectrum  of  the  metal.     Metals, 
such  as  sodium,  or  salts  of  such  metals,  can  be  introduced  into 
a  flame  supplied  with  oxygen.     The  chemical  reactions  which 
take  place  in  the  hot  flame    cause    the  vapor  to  chemically 

1  Fredenhagen,  Phys.  Zeit.,  1907,  p.  404,  p.  729. 


r. 


KIRCHOFF'S  LAW.  207 

luminesce,  and  emit  its  "line"  spectrum.  Vapors  of  iodine 
luminesce  where  there  are  great  changes  of  temperature  or 
pressure,  and  give  the  characteristic  spectrum  of  iodine. 
Such  luminiscence  is  probably  due  to  recurring  dissociation 
and  association.1 

The  radiations  from  a  gas  or  vapor  are  usually  confined 
to  such  exceedingly  limited  groups  of  waves  that  while  a 
por  such  as  iron  may  emit  several  thousand  such  groups,  the 
higher  the  resolving  power  of  the   spectrometer,   the  more 
isolated  appear  the  wave  lengths  in  each  individual  group. 
The  great  delicacy  of  spectrum  analysis  is  due  to  the  fact 
hat  different  gases  or  vapors  emit  waves  of  different  wave 
gth,  and  an  exceedingly  small  amount  of  gas  gives  intense 
diations  under  electric  luminiscence. 

217.  Kirchoffs  Law. — Although  Kirchoff's  law  of  propor- 
ionality  between   absorption   and   emission   has   only  been 
ved  for  pure  thermal  radiation,  and  although  the  emission 
m  vapors  and  gases  appears  to  be  due  largely  to  luminiscence, 
et  gases  and  vapors  generally  absorb  the  radiations  which 
ey  emit.     This  principle  has  often  been  used  as  a  method  of 
alysis,  in  cases  where  the  electric  discharge  required  for  an 
ission  spectrum,  would  be  likely  to  change  the  composition, 
e  gas,  or  vapor,  is  inclosed  in  a  long  tube  through  which  the 
ight  from  an  incandescent  source  passes,  before  entering  a 
trometer.     The  spectrum  will  be  crossed  by  more  or  less 
w  bands   of   all   degrees   of  intensity.     Warburg2   has 
ently   used   this  method   in   a   beautiful   qualitative   and 
uantitative  analysis  of  the  various  oxides  of  nitrogen,  pro- 
uced  by  electric  discharge.     He  worked  with  waves  longer 
an  those  which  affect  the  eye  (infra-red)  and  measured  the 
tensity  of  the  radiations  with  a  very  narrow  bolometer  (§48), 
hich  was  placed  successively  in  different  parts  of  the  spectrum. 
Such    bands    of   appreciable    width    are    characteristic    of 
mpounds,  as  the  narrow  lines,  described  in  the  previous 
ragraph  are  characteristic  of  the  elements. 

1  Fredenhagen,  /.  c. 

2  Ann.  der  Phys.,  1909,  xxviii,  p.  313. 


208  LIGHT. 

218.  Absorption  Spectrum  of  Liquids. — Comparatively  few 
liquids  retain  their  characteristic  properties  at  temperatures 
which  are  high  enough  to  afford  appreciable  radiation.  We 
are  therefore  obliged  to  study  the  degree  to  which  they  absorb 
the  light  from  an  incandescent  solid.  All  liquids  show  some 
absorption,  but  it  is  generally  extended  over  quite  a  range  of 
wave  lengths,  and  the  transition  between  absorption  and 
comparative  transparency  is  usually  gradual.  Therefore, 
in  the  case  of  liquids,  the  absorption  spectrum  is  not  generally 
a  delicate  means  of  analysis. 

Solutions  of  the  rare  earths,  erbium,  didymium,  europium, 
and  neodymium,  absorb  limited  groups  of  waves,  and  therefore, 
when  light  transmitted  through  such  solutions  is  examined 
with  a  spectrometer,  the  spectrum  is  crossed  by  dark  bands. 
Potassium  permanganate,  in  a  thin  layer,  of  dilute  solution, 
shows  five  dark  bands  in  the  yellow  and  green.  All  other 
.permanganates  show  the  same  identical  bands,  and  it  is  there- 
fore concluded  that  the  absorption  is  due  to  the  negative, 
permanganate  ion  (§168).  The  narrowing  of  absorption 
bands  with  dilution  is  explained  by  Jones1  as  due  to  increasing 
hydration  (§304)  of  the  ions.  The  more  an  ion  is  loaded  with 
water  molecules,  the  more  limited  is  the  number  of  vibrations 
to  which  it  can  respond  and  which  it  can  thereby  absorb. 
Numerous  other  salts  which  have  a  colored  ion  (copper  ion  = 
blue,  cobalt  ion=red,  ferrous  ion=green,  ferric  ion=yellow, 
etc.)  give  more  or  less  distinctive  absorption  bands,  details  of 
which  are  given  in  the  references.2 

TABLE  XXX. 

Wave  Lengths  of  Ether  Waves  (mm.). 
Shortest  waves  investigated oooi 


Shortest  waves  ordinarily  transmitted  by  air 

Shortest  visible  waves  (violet) 

Blue 

Green 

Yellow  . . 


.00018 
.00038 
.00045 
.00052 
.00057 

Longest  visible  waves  (red) 0007  5 

Longest  waves  measured  by  heat  effect 0612 

Shortest  waves  measured  by  electric  effect ...    6.0 

1  "Elements  of  Physical  Chemistry,"  p.  242. 

2  See  Carnegie  Institution  publications  (60.  no)  of  Jones  and  Uhler. 


SPECTROSCOPY.  209 

EXPERIMENT  XXVI. 
Spectroscopy.     Emission  and  Absorption  Spectra. 

The  construction  and  adjustments  of  the  spectrometer  (§61) 
must  be  studied  carefully.  The  grating  spectrometer  (§64)  is  prefer- 
able, since  it  measures  wave  lengths  directly. 

(A)  Emission  Spectrum. — The  assigned  spectrum  tube1  should  be 
mounted  in  front  of  the  slit  of  the  spectrometer  so  that  the  capillary, 
in  which  the  light  is  concentrated,   coincides  with  the  axis  of  the 
collimator.     The  terminals  are  connected  to  a  small  induction  coil, 
capable  of  giving  a  spark  of  at  least    i   cm.     The  slit  is  made  as 
narrow  as  is  consistent  with  sufficient  illumination.      The  cross  hair 
of  the  telescope  is  set  successively  on  each  of  the  more  conspicuous 
bright  lines  in  the   spectrum  and  the  position  of  the  telescope  is 
read  on  the  graduated  circle.     After  each  prominent  bright  line  in 
the  spectrum  has  been  located  on  both  sides  of  the  undeviated  ray, 
a  second  set  of  readings  is  made  on  each  side,  and  finally  a  third  set, 
and   the    mean   position   of   each   line   is    calculated.      The   angular 
deviation  of  each  line  is  half  the  angle  between  the  positions  on  the 
two  sides.      If  a  grating  is  used,  and  some  other  spectrum  than  the 
first  is  brightest,  this  spectrum  should  be  used  and  the  proper  value 
of  the  order,  n,  substituted  in  the  formula  (25) 

a  sin  6  ^ 

n 

Tabulate  the  wave  lengths  and  colors  of  the  most  prominent  lines. 

(B)  Absorption     Spectra. — The     assigned     liquid2     is     contained 
in  a  hollow  glass  prism  which  is  mounted  horizontally,  in  front  of 
the  slit  of  the  spectrometer.     An  incandescent  lamp,  with  a  ground- 
glass  bulb,  is  a  convenient  source  of  light.     This  is  placed  as  close 
as  possible  to  the  prism  and  a  little  below  the  level  of  the  slit,  on 
account  of  the  refraction  of  the  prism.      It  is  properly  shielded  to 
prevent  any  light  entering  the  spectrometer  without  traversing  the 
prism.     The  lower  half  of  the  slit  is  therefore  illuminated  with  light 
which  has  traversed  a  considerable  thickness  of  solution,  while  the 
light  which  enters  the  upper  part  of  the  slit  has  passed  through  very 
little    of   the    solution.     Therefore,    a    comparison    of   the    top    and 
bottom  of  the  spectrum  will  show  the  effect  of  increasing  the  thickness 
of  the  solution. 

Set  the  cross  hair  of  the  telescope  successively  upon  the  center  of 
each  absorption  band  on  each  side,  and  read  its  position.  Repeat 
the  readings  at  least  twice,  and  find  the  mean  angular  deviation, 
d,  of  each  band  (half  the  angle  between  the  positions  on  the  two 
sides),  and  from  this,  the  wave  length.  Tabulate  the  wave  lengths 
and  the  approximate,  absorbed  color. 

1  Hydrogen,  water  vapor,  ammonia,  carbon  dioxide,  and  carbon  monoxide 
give  satisfactory  spectra.     Excellent  tubes  may  be  obtained  of  Goetze,  Leipsic, 
or  Muller-Uri,  Braunschweig. 

2  It  is  interesting  to  have  different  students  find  the  positions  of  the  absorp- 
tion bands  of  different  permanganates,  e.g.,  sodium,  potassium,  calcium,  etc., 
and  compare  the  results,  when  the  experiment  has  been  completed  by  all. 

14 


210  LIGHT. 

QUESTIONS. 

1.  How  would  the  appearance  of  the  emission  spectrum  depend 
upon  (a)  the  pressure  of  the  gas?      (6)  the  current  through  the  tube? 
(c)  the  direction  of  the  current?      (d)  the  length  of  the  capillary? 

2.  (a)  Did  the  top  of  the  spectrum  correspond  to  a  thick  or  narrow 
layer  of  solution  ?     (&)  For  which  were  the  absorption  bands  narrow- 
est?    Explain. 

219.  Velocity  of  Light.  Refractive  Index.  —  Light  travels 
in  pure  ether  with  a  velocity  of  3  Xio10  cm.  per  sec.  In  all 
transparent  material  bodies  the  velocity  is  less,  but  the  differ- 
ence is  hardly  appreciable  for  gases.  (For  air,  under  normal 
conditions,  the  ratio  is  1.000293.)  The  ratio  of  the  velocity 
in  ether  (or  air  unless  extreme  accuracy  is  required)  to  the 
velocity  in  the  body,  is  called  the  refractive  index  of  the  body. 
We  will  designate  this  quantity  by  /*.  It  is  shown  in  physics 
text-books  that  when  a  ray  of  light  (direction  of  advance  of 
waves)  passes  obliquely  from  air  (or  ether)  into  a  body  of 
refractive  index  JJL 

sm<f)I  . 

'""  (I34) 


where  </>x  is  the  angle  between  the  ray  in  the  air  and  the  normal 
to  the  surface,  and  (f>2  is  the  angle  between  the  normal  and 
the  ray  in  the  body.  </>j  is  greater  than  <£2>  and,  if  the  ray 
passes  from  the  body  to  air,  0Z  is  equal  to  90°  for  a  particular 
value  of  (j)2  (called  the  critical  angle,  </>c)  . 


If  the  light  passes  from  a  body  whose  refractive  index  is 
to  a  body  where  refractive  index  is  jj.2 


(I36 


For,  if  Ue  is  the  velocity  in  ether  and  t7x  and  U2  are  the  velocities 
in  the  two  bodies 

Ue  Ue 

^  =  T7T'  ^=U7 

/*!  _  U2  sin  <f>2 

H2      Uj.  sin  <t 


PULFRICH    REFRACTOMETER. 


211 


If  a  ray  of  light  passes  through  a  prism  whose  angle  is  A  and 
whose  refractive  index  is  /*,  the  minimum  angle  of  deviation  of 
the  ray,  D,  is  given  by  the  equation 


sin 


A  +  B 


(i37) 


sin 


The  most  accurate  method  of  determining  the  refractive 
index  of  a  solid  is  to  cut  it  into  the  form  of  a  prism  and 
measure  the  angle  of  the  prism  and  the  angle  of  minimum 
deviation,  and  calculate  JJL  by  Equation  137.  Liquids  may  be 
inclosed  in  a  hollow  prism  with  sides  of  truly  plane  glass.  A 
more  convenient  method  of  measuring  the  refractive  index  is 
to  find  the  critical  angle  between  the  body  and  another  body 
whose  refractive  index  is  known,  and  calculate  /*  by  Equations 
135  and  136.  Such  an  instrument  is  called  a  refractometer. 

220.  Pulfrich  Refractometer. — Fig.  74  illustrates  the  prin- 
ciple of  the  Pulfrich  refractometer.  The  liquid  is  contained 
in  a  glass  cell  which  rests  on  the  horizontal 
face  of  a  right-angle  prism.  A  horizontal 
beam  of  light ,  from  a  monochromatic  source 
(§60),  is  focused  by  a  lens,  upon  a  point  in 
the  surface  between  the  liquid  and  the  glass. 
A  horizontal  beam  of  light  will  enter  the 
prism  at  the  critical  angle,  e  (see  figure), 
and  will  emerge  from  the  prism  at  the  angle  -  74' 

i,  which  is  observed  with  a  telescope.     If  N  is  the  refractive 
index  of  the  prism,  and  JJL  that  of  the  liquid, 


sin  i 


sin 


sin  ^ 


sin  (90°  —  e)      cos  e 


for 


sin  e 


N 


(Equation  136) 


=\/N2  —  sin2  i 


212 


LIGHT. 


Values  of  N  for  different  wave  lengths  (colors)  are  furnished 
with  the  instrument.  It  is  well  to  test  the  instrument  first, 
with  a  liquid  whose  refractive  index  has  been  carefully 

determined  (see  Table  XXXI). 
Directions  for  obtaining  the  zero 
correction  are  given  under  Experi- 
ment XXVII. 

Instruments  are  often  furnished 
with  a  closed  tube  which  stands  in 
the  top  of  the  liquid  and  through 
which  passes  a  constant  stream  of 
water  from  a  thermostat,  which 
serves  to  keep  the  temperature  of 
the  liquid  constant. 

221.  Abbe  Refractometer. — Fig. 
75  represents  a  form  of  Abbe  re- 
fractometer  made  by  Zeiss,  of 
Jena,  which  is  very  satisfactory 
for  general  laboratory  use.  A  very 
thin  layer  of  the  liquid  is  inclosed 
between  two  prisms  (Fig.  79)  and 
the  angle  i  is  determined  from  the 
difference  in  the  position  of  the 
prisms  P  (Fig.  75),  when  light 
traverses  the  liquid  at  grazing  in- 
cidence, and  when  i  is  zero.  The 
latter  position  is  obtained  when  the 
reflection  of  a  portion  of  the  cross 
hairs  which  is  illuminated  by  a 
small  prism  (a),  Fig.  75,  coincides 
with  the  cross  hairs  on  the  other 
side  (see  Fig.  76).  The  position  of  the  prism  for  the  grazing 
ray  illustrated  in  Fig.  79  is  the  position  where  the  cross 
hairs  meet  on  the  dividing  line  between  darkness  and  light. 
Monochromatic  light  is  necessary  for  accurate  setting.  The 
position  of  the  prism  is  read  by  means  of  a  vernier  reading  to 
two  minutes,  attached  to  the  arm  A . 


FIG.  75. 


ABBE    REFRACTOMETER. 


213 


If  /t  is  the  refractive  index  of  the  film  of  liquid  and  N  that 
of  the  glass 

fi  =  N  sin  e  (Equations  135  and  136,  and  Fig.  79)       (139) 


FIG.  76. 


We  must  determine  e  in  terms  of  ^  the  angle  of  the  prism 
and  i  the  angle  observed.     We  have 


e  =  ()  —  r 
sin  i 


sin  r=- 


N 


(140) 
(141) 


If  </>  and  N  are  not  given,  they  can  easily  be  determined. 
</>  is  measured  by  finding  the  positions  of  the   single   upper 


FIG.  78 


FIG.  79. 


prism,  for  reflection  of  the  illuminated  cross  hairs,  from  each  of 
the  two  faces  inclosing  the  angle.     The  angle  between  these 
two   positions  is  evidently  the   supplement  of  <j>  (§62). 
After  (j>  has  been  determined,  N  is  determined  by  the  above 


214 


LIGHT. 


equations,  from  the  value  of  i  for 
139,  140  and  141 


i',  that  is,  for  air.     From 


sin2        sn 


/ 
-sin2  *= 


A^  ----  ATC°S 

i  +  sin  i  cos  <i 

—  -^  —  :  —  -7 
sin 


/ 
(141 


*  should  be  taken  as  the  mean  of  the  values  observed  for  the 
two  rays  illustrated  in  Figs.  77  and  78. 

222.  Refractivity.     The   refractivity  or  refractive  power  is 
defined  by  the  expression1 


I     fJL2  —  I 

p  p2  +  2 

223.  The  molecular  refractivity  is 


(142) 


M  //2-  i 

~ 


(143) 


Where  M/p  is  the  molecular  volume  (§ 


TABLE   XXXI. 


Refractive  Indices  (Yellow  Light  (D  lines)  20°). 

Acetone 

Ethyl  ether 

Ethyl  alcohol 

Aniline 

Benzol 

Chloroform 

Acetic  acid 

Glycerine 

Methyl  acetate    

Ethyl  acetate 

Pyridine 

Toluene 

Water 

Carbon  bisulphide 


•36138 
.58629 
.50144 

•449° 
.37182 

•47293 
.30099 

•37257 
.50880 

•49552 

•3329 

.6277 


1  Lorentz,  Wied.  Ann.,  1880,  ix,  p.  641;  Lorenz,  Wied.  Ann.,  1880,  xi,  p.  70. 


ATOMIC    REFRACTIVITIES. 


2I5 


TABLE  XXXII. 
Atomic  Refractivities  (Briihl1  and  Conrady2). 


Element 

Symbol 

Red  (Ho) 

Yellow  (D) 

Blue  (H^) 

Hydrogen     

H 

I  .  I  O  "? 

i  o  si 

I.IT.Q 

Chlorine  

Cl 

6014 

r  008 

6.  loo 

Bromine              

Br 

8  863 

8  027 

92  I  I 

Iodine 

I 

it  808 

14.   12 

14..  ^82 

Oxygen   (OH  group)  .  .  . 

0' 

1.506 

I.52I 

I-525 

Oxygen  (in  ethers)  

o< 

MSS 

1.683 

1.667 

Oxygen  (in  CO  group) 

O" 

2.328 

2.287 

2.414 

Carbon  

c 

2.36  1; 

2.  SOI 

2.404 

Ethylene  bond  



i.  8^6 

I.7O7 

i.Sso 

Acetylene  bond  

— 

2.27 

2.IO 

2-40 

The  atomic  refractivity  is  similarly 


A  fJL*  -  I 
p 


+2 


(M4) 


where  A  is  the  atomic  weight. 

Exners  has  shown  that  the  actual  volume  occupied  by  the 
molecules  may  be  estimated  from  the  molecular  refractivity. 
The  values  so  obtained  are  less  than  Van  der  Waal's  constant 
6  (§120).  That  the  molecular  refractivity  is  an  additive 
property  for  many  solutions  has  already  been  mentioned 
(§1676).  The  atomic  refractivity  is  an  additive  property  for 
many  organic  liquids.  If  G  is  the  molecular  refractivity  of 

such  a  liquid  and  gl,  g2, are  the  atomic  refractivities 

of  the  atoms  whose  numbers  are  respectively  nlt  n2 

G  =  nIgI  +  n2g2  +  -  (145) 

1  Zeit.  phys.  Chem.,  1889,  iii,  p.  226. 

2  Zeit.  phys.  Chem.,  1891,  vii,  p.  146. 

3  Wien.  Monatshefte  f.  Chemie,  1885,  vi,  p.  249. 


2 1 6  LIGHT. 

Table  XXXII  gives  Bruhl  and  Conrady's  values  of  the  atomic 
refractivities  of  the  more  common  elements  in  organic  liquids. 

Illustration. — The  calculated  molecular  refractivity  of  benzol 
(C6//6),for  red  light,  is  6X2.365+6X1.103+3X1.836=26.32. 
Direct  experiment  gave  /*  =  1.501 44,  />  =  .88oandM  =  78. 
Substituting  in  Equation  143,  £  =  26.13 

224.  Dispersion. — The  refractive  index,  and  therefore  the 
refractivity,  varies  with  the  wave  length  (or  color)  of  the  light. 
The  difference  in  the  refractive  indices  for  two  colors  is  called 
the  dispersion  for  these  colors,  and  the  difference  in  refrac- 
tivities is  called  the  dispersivity.  The  molecular  and  atomic 
dispersivities  are  analogous  to  the  molecular  and  atomic 
refractivities. 

EXPERIMENT  XXVII. 

Refractive  Indices  and  Refractivities  of  Liquids.     Use  of 
Refractometer. 

First  study  carefully  the  construction  and  use  of  the  refractometer. 
All  the  glass  surfaces  must  be  made  scrupulously  clean.  A  very 
soft  clean  cloth  should  be  used.  If  a  Pulfrich  instrument  is  to  be 
employed,  determine  the  zero;  that  is,  the  position  of  the  telescope 
when  the  axis  of  the  telescope  is  parallel  to  the  top  of  the  prism. 
A  portion  of  the  cross  hairs  is  illuminated  by  a  small  prism  (Fig.  76). 
The  telescope  is  adjusted  until  the  image  of  this  portion  of  the  cross 
hairs,  reflected  from  the  vertical  side  of  the  prism,  coincides  with 
the  cross  hairs  on  the  other  side.  The  refractive  index  and  the 
angle  of  the  prism  of  an  Abbe  refractometer  must  be  determined  as 
described  in  §221. 

(A)  Determine   the   refractive    index   of   pure   water   and    of   an 
assigned  solution.      Determine  also  the  density  of  the  latter  (§36). 
The  density  of  water  at  the  temperature  'of  the  experiment   may 
be  obtained  from  Table  LI.     Calculate  the  refractivity  of  water  and 
of  the  solution,  and,  finally,  by  Equation  109,  determine  the  refrac- 
tivity of  the  dissolved  substance.    Monochromatic  light  (§60)  should 
be  used.     If  possible,  use  several  colors,  but  if  only  one  monochro- 
matic source  is  available,  make  one  determination  with  white  light 
and  find  the  approximate  angle  for  both  extremes  of  the  spectrum, 
The  difference  in  refractive  indices  (dispersion)  should  be  calculated, 
and  also  the  difference  in  refractivities  (dispersivity). 

(B)  Determine  the  refractive  index  and  dispersion  of  an  assigned 
organic  liquid  and  also  the  density  (§36),  unless  it  is  given  in  Table 
LIV.     Calculate  its  refractivity  and  molecular  refractivity  and  com- 
pare with  the  value  calculated  from  the  atomic  refractivities  (Table 
XXXII).      Estimate    the     dispersion,    dispersivity    and     molecular 
dispersivity. 

(C)  If  time  permit,  determine  the  refractive  indices  of  a  series  of 
normal  or  half-normal  solutions  similar  to  those  used  in  Experiment 
XVI,  and  demonstrate  that  the  refractive  index  is  an  approximate 
additive  property  for  highly  dissociated  solutions. 


ROTATORY    POLARIZATION.  217 

QUESTIONS. 

1.  What  is  the  velocity  of  light  (a)  in  the  first  liquid?     (6)  in  the 
second? 

2.  A  ray  of  light  strikes  the  liquid  surface  at  an  angle  of  incidence 
of   30°.      What   is   the   angle   of  refraction   in    (a)    the   first   liquid? 
(6)  the  second? 

3.  What  is  the  critical  angle  of  (a)  the  first  liquid?     (6)  the  second? 

4.  If  the  second  liquid  forms  a  layer  upon  the  first,  and  a  ray  of 
light   meets  the  boundary  at  an  angle  of  incidence  of  45°  in  the 
first  liquid,  what  is  the  angle  in  the  second? 

Rotatory  Polarization. 

225.  The  vibrations  of  ordinary  light  are  in  every  direction 
and  of  every  conceivable  form.  Certain  instruments,  particu- 
larly the  Nichol's  prism,  resolve  these  heterogeneous  vibra- 
tions into  linear  vibrations  in  one  direction  and  the  light  is  then 
said  to  be  plane-polarized.  Certain  bodies  have  the  property 
of  changing  the  direction  of  vibration  of  such  light;  that  is, 
as  the  plane-polarized  light  advances  through  the  medium,  the 
direction  of  vibration  continuously  rotates.  This  phenomenon 
is  called  rotatory  polarization,  and  the  bodies  are  said  to  be 
optically  active. 

Optically  active  bodies  have  a  type  of  dissymmetry  il- 
lustrated by  a  screw  or  helix.  For  a  given  direction  of  ad- 
vance, the  structure  is  different  for  one  direction  of  rotation 
from  what  it  is  for  the  other.  Just  as  there  are  right-  and 
left-handed  screws,  so  there  are  right-  and  left-handed  opti- 
cally active  bodies.  If  the  direction  of  rotation  appears 
clockwise  to  an  observer  receiving  the  light,  it  is  called  right- 
handed,  and  vice  versa. 

A  quartz  crystal  has  such  a  dissymmetry  in  the  arrangement 
of  its  faces,  and,  according  to  the  form  of  the  dissymmetry,  it 
rotates  the  plane  of  polarization  to  the  right  or  left.  A 
similar  condition  exists  among  the  atoms  of  a  molecule  having 
an  asymmetric  carbon  atom;  for  example,  dextro-  or  laevo- 
tartaric  acid.  When  a  transparent  body  is  subjected  to  twist 
about  an. axis  coincident  with  the  direction  of  the  light,  the 
mechanical  structure  acquires  similar  dissymmetry  and  the 
body  shows  rotatory  polarization.1 

1  Ewell,  Am.  Jo.  of  Science,  1899,  viii,  p.  89;  1903,  xv,  p.  363;  Phys.  Zeit., 
1899,  p.  18;  1903,  p.  706;  Johns  Hopkins  Circulars,  1900,  June. 


2l8  LIGHT. 

In  all  these  cases  the  structure  is  such  that  a  light  vibration, 
consisting  of  rotation  in  a  circle  about  the  direction  of  advance  of  the 
light,  travels  with  different  velocities  according  as  the  direction  of 
rotation  agrees  with,  or  is  opposite  to,  a  direction  which  is  determined 
by  the  structure.  It  is  easily  shown  graphically  and  analytically 
that  a  linear  vibration  is  equivalent  to  two  opposite  circular  vibra- 
tions, and  if  one  of  these  travels  faster  than  the  other,  the  direction 
of  the  resultant  linear  vibration  rotates. 

226.  In  physical  chemistry  we  are  particularly  concerned 
with  the  optical  activity  of  bodies  having  one  or  more  assym- 
metric  atoms  and  particularly  with  solutions  of  such  bodies. 
The  amount  of  rotation,  0,  depends  upon  the  length  of  the  solu- 
tion, the  mass  of  solute  in  unit  volume  of  the  solution,  the 
temperature,  and  the  wave  length  of  the  light  employed. 
If  m  =  mass  of  solute,  m0  =mass  of  solvent,  /  =length  of  solu- 
tion, and  p=  density  of  the  solution,  the  volume  of  the  solu- 
tion is  (m  +m0)  /  p  and  the  mass  of  solute  in  unit  volume  of  the 
solution  is 

mp 


-  (i46) 

The  constant  [<£]*£>  is  called  the  specific  rotatory  power 
or  specific  rotation.  The  lower  suffix  (D)  states  the  kind  of 
light  employed  and  the  upper  (t) ,  the  temperature.  /  is  usually 
expressed  in  decimeters.  If 

/=i,  and  •_  mfj '     =i, 


or,  the  specific  rotatory  power  is  the  rotation  produced  by  one 
decimeter  of  solution  which  contains  one  gram  of  active  sub- 
stance per  cubic  centimeter.  For  a  pure  active  liquid,  m0  =  O 
and 


(i47) 


MAGNETIC    ROTATION. 


219 


The  specific  rotatory  power, 'multiplied  by  the  molecular  weight, 
r,  is  called  the  molecular  rotatory  power .     For  convenience,  one 
lundredth  of  this  constant  is  commonly  employed. 

227.  For  many  substances,  such  as,  for  example,  the  sugars, 
the  specific  rotatory  power  is  approximately  constant ;  that  is, 
is  quite  independent  of  the  concentration,  and  therefore 
:he  rotation  of  the  plane  of  polarization  may  be  used  as 
convenient  method  of  measuring  the  concentration.     With 
lany  other  substances,  however,  it  varies  with  the  concentra- 
tion, and,  if  dissolved,  with  the  nature  of  the  solvent.     If, 
lowever,  different  salts  of  an  active  base  or  of  an  active  acid 
ire  dissolved  in  a  common  solvent,  and  a  number  of  decreasing 
mcentrations  of  the  resulting  solutions  are  examined,  the 
>tation  at  great  dilution,   for  all  the  salts,   has  the  same 
ralue  (Oudeman's  law).     This   obviously  is   in  accord   with 
ie  dissociation  theory  ( §  1 68) . 

TABLE  XXXIII. 

Specific  Rotatory  Power  (20°).    Yellow  Sodium  Light  (D).* 


Active  Substance 

Concentration 
(=c)  (gr.in  TOOC.C.) 

[0]2C 

>  

^ane-sugar,  R  

/    3-28 
\  10-86 

66.639  —  . 
66.453-. 

0208(7 

000124^ 

nvert  sugar,  L  

1-14 

—  20.07    ~- 

041(7 

Glucose    (dextrose).    R 

(crystallized)  

0-100% 

4773     ~\~ 

01  5  X  % 

7ructose  (levulose)  L  .  . 

0-40 

*T  I        1    J 

—  100-3        +• 

Milk-sugar,  R  

5-7 

5^-53 

Fartaric  acid,  R  
Quartz,  R  or  L  

I  22-63 

15.06    - 
13.436-. 
2  1.  70  (tor  i  mm 

I$IC 

ii  gc 
.  thickness) 

228.  Magnetic  Rotation. — It  is  explained  in  treatises  on 
physical  optics  that  a  certain  rotation  is  associated  with  a 
magnetic  field,  which  in  many  bodies  rotates  the  direction  of 
the  light  vibrations  of  a  ray  of  light  parallel  to  the  magnetic 
field. 

There  is,  however,  no  such  dissymmetry  associated  with 

1  Landolt  and  Bornstein. 


220 


LIGHT. 


this  rotation  as  is  illustrated  by  a  screw.  The  light  vibra- 
tions are  rotated  in  the  same  direction,  whether  the  ray 
is  traveling  in  the  direction  of  the  magnetic  field  or  in  the 
opposite  direction.  The  amount  of  the  rotation  is  propor- 
tional to  the  integral  product  of  the  strength  of  the  magnetic 
field  and  the  length  of  the  body. 

The  magnetic  rotatory  power  of  a  substance  may  be  defined 
in  either  of  two  ways.  The  absolute  unit  is  called  Verdet's 
constant  and  is  equal  to  the  rotation,  in  minutes,  of  a  column 
one  centimeter  long  in  unit  (C,  G,  5.)  magnetic  field  (§8). 
Often,  however,  the  rotatory  power  is  measured  by  the  ratio 
of  the  rotation  in  the  substance  to  that  in  a  column  of  water, 
of  equal  length,  in  a  similar  magnetic  field.  This  ratio,  divided 
by  the  density  of  the  substance,  is  called  the  specific  magnetic 
rotation. 

229.  Rotatory  Dispersion. — Both  the  natural  and  the 
magnetic  rotation  of  the  plane  of  polarization  are,  approxi- 
mately, inversely  proportional  to  the  square  of  the  wave 
length  of  the  light  employed. 

TABLE   XXXIV. 

Magneto-optic  Rotation  (20°).     Yellow  Sodium  Light  (D). 


Liquid 

Verdet's  Constant  in 
Minutes  l 

Specific  Magnetic 
Rotation 

Acetone  
Amyl  alcohol  
Ethyl  alcohol  
Methyl  alcohol  
Benzol  •. 

.0113' 
.0128 

.OI  12 
.0093 

1.080 
1.204 
1.070 
9i3 

2     ^O  2 

Carbon  bisulphide  
Chloroform  
Toluene  
Water  

.0441 
.0164 
.025 
.0132 

2-505 
.839 

2-354 

i 

230.  Polarimeters. — The  simplest  polarimeter  consists  of 
a  Nichol's  prism  (polarizer)  to  resolve  the  light  vibrations  into 
one  direction,  a  tube  to  contain  the  specimen,  and  a  second 
Nichol's  prism  (analyzer)  which  locates  the  direction  of  vibra- 

1  Smithsonian  Tables,  No.  305. 


POLARIMETERS. 


221 


tion  of  the  light  after  it  has  traversed  the  specimen.  The 
inalyzer  is  set  so  as  to  transmit  no  light.  Such  is  evidently 
the  case  when  the  direction  of  vibration  of  the  light  incident 
>n  the  analyzer  is  perpendicular  to  the  vibrations  which  it 
would  transmit.  The  specimen  is  then  removed  and  the  analy- 
zer is  reset  for  darkness.  The  angle 
between  the  two  positions  is  obviously 
equal  to  the  rotation  of  the  specimen. 
The. setting  of  a  Nichol's  prism  for 
darkness  is  usually  uncertain  by  one 
or  two  degrees,  and  therefore  more 
accurate  instruments  have  been  de- 
vised. : 

231.  We  will  first  describe  the  bi- 
quartz  polarimeter.     The  biquartz  is  a   E 
double  plate,  one  side  being  of  right- 
handed  quartz  and  the  other  of  left- 
handed     quartz,    and    the    simplest 
mounting  is  illustrated  in  Fig.  81. 
The  analyzer  is  rotated  until  both 
lives   of   the   biquartz    appear    the  FlG-  8o- 

tme  (purple)  color,  first  without  the  specimen,  then  with 
the  specimen ;  and  the  difference  in  readings  gives  the  angle  of 
rotation  for  yellow  light.  There  are  additional  collimating 
lenses  which  render  the  light  parallel  before  entering  the  solu- 
tion, and  observing  lenses  which  give  a  clear  image  of  the  bi- 
luartz.  Fig.  80  explains  the  color  changes.  R  and  L 


FIG.  81. 

are  the  two  halves  of  the  biquartz,  viewed  from  the  analyzer. 

'he  two  halves  are  of  such  a  thickness  (3.75  mm.)  that  the 
plane  of  polarization  of  yellow  light  is  rotated  through  90°. 

)wing  to  the  rotatory  dispersion  (§229),  the  other  colors  will 
be  rotated  different  amounts  as  shown  by  the  letters  R 


222  ...       LIGHT. 

(red)  and  B  (blue).  If  the  analyzer  is  set  to  transmit  light 
vibrations  parallel  to  those  which  left  the  polarizer,  the  yellow 
light  will  be  omitted  and  each  half  of  the  biquartz  will  appear 
of  a  purplish  color  ("tint  of  passage").  If  the  analyzer  is 
displaced  slightly  clockwise,  more  of  the  red  component  on 
the  right  will  be  transmitted  and  less  of  the  blue,  and  there- 
fore this  half  will  appear  red  and  the  other  half  will  appear 
blue. 

If  a  dextro-rotatory  specimen  is  placed  between  the  bi- 
quartz and  the  analyzer,  the  directions  of  vibration  of  the 
different  colors,  will  be  rotated  to  the  positions  indicated  by 
the  dotted  lines  and  the  analyzer  must  be  rotated  to  a  new 
position  (/'),  perpendicular  to  the  emerging  yellow  vibration, 
in  order  to  have  the  two  halves  the  same  color.  With  the 
help  of  the  biquartz  the  analyzer  can  be  set  within  about  a 
tenth  of  a  degree. 

232.  The  essential  parts  of  the  Lippich  type  of  half-shade 
polarimeter  are  illustrated  in  Fig.  82.  The  smaller  Nichol's 
prism  of  the  polarizer  is  at  a  small  angle  with  the  larger 


FIG.  82. 

Nichol's  prism,  so  that,  unless  the  analyzer  is  properly  placed, 
bisecting  the  angle  between  them,  the  two  parts  of  the  field 
appear  of  different  shades.  The  analyzer  is  set  so  that  both 
halves  of  the  field  appear  equally  bright,  first  without  the 
specimen,  then  with  the  specimen,  and  the  difference  is  the 
angle  of  rotation. 

The  angle  between  the  two  Nichol's  prisms  of  the  polarizer 
can  be  varied.  The  smaller  the  angle,  the  greater  is  the 
sensitiveness,  but  the  illumination  is  thereby  reduced.  There 
are  also  lenses  for  making  the  incident  light  parallel  and  for 
observing  the  line  of  separation  of  the  compound  polarizer. 
The  addition  of  the  half- shade  device  gives  greater  sensitive- 
ness than  the  biquartz  (about  .01  degree). 


SACCHARIMETERS. 


223 


In  all  polarimeters  there  are  obviously  two  positions  of 
the  analyzer,  180°  apart,  where  the  two  halves  appear  equally 
dark  or  the  same  dark  color.  There  are  also  two  intermediate 
positions  where  the  two  halves  appear  the  same  bright  color  or 
equally  bright.  The  latter  positions  are  not  as  sensitive  as 
the  former. 

Since  the  rotatory  power  of  most  substances  changes  with 
the  temperature,  in  accurate  work,  the  observation  tube  should 
be  surrounded  by  a  jacket  through  which  flows  water  from  a 
thermostat. 

233.  Saccharimeters. — With  some  polarimeters,  particularly 
those  used  for  sugar  analysis  (saccharimeters) ,  the  analyzer 


FIG.  83. 

is  not  rotated,  but  the  rotation  produced  by  the  specimen  is 
neutralized  by  a  quartz  wedge  or  wedges  (see  Fig.  83).  The 
letters  specify  the  rotations  of  the  quartz.  If  the  wedges  are 
drawn  apart,  right-handed  rotation  will  be  neutralized,  and 
vice  versa.  A  scale  is  attached  to  one  wedge  and  the  unit  is 
usually  the  Ventske,  i.e.,  100  divisions  equal  the  rotation 
produced  by  26.048  gr.  of  sugar  in  100  c.c.  of  solution,  at 
17.5°,  in  a  tube  20  cm.  long  (i  mm.  of  quartz  produces  the 
same  rotation) . 

PROBLEMS  XII. 

1.  (a).  Calculate  the  amount  of  energy  radiated  in  one  hour  by  a 
carbon  sphere   i   cm.  in  diameter  at  a  temperature  of  1000°.      (6) 
What  is  the  wave  length  of  the  most  intense  radiation  from  such  a 
source  ? 

2.  Langley  found  the  most  intense  radiations  from  the  sun  at 
wave   length   .0005   mm.     Calculate   the   approximate  temperature 
of  the  sun. 

3.  (a)   At  any  instant  how  many  red  waves  are  comprised  in  a 
distance   of   one   metre   from   a   hydrogen   tube   in   operation?      (6) 
How  many  are  issued  per  second  ?    [Remember  that  u  =nX,  (Eq.  88'")]. 

4.  What   is  the    velocity   of   yellow   light    (a)    in   water?      (6)    in 
carbon  bisulphide? 


224  LIGHT. 

5.  A  layer  of  water  is  poured  upon  carbon  bisulphide.     A  ray  of 
light  enters  the  water  at  an  angle  of  30°  in  air.      What  is  the  angle 
in  (a)  the  water?      (b)  in  the  carbon  bisulphide? 

6.  Calculate  the  molecular  refractivity  of  water  (D  light)  and  com- 
pare the  result  with  that  calculated  from  the  atomic  refractivities. 

7.  Calculate  the  rotation  produced  by  50  cm.  of  (a)  a  2%  solution 
of  invert  sugar;  (6)  a  2  %  solution  of  dextro-tartaric  acid. 

8.  What  is  the  concentration  of  a  sugar  solution,  if  a  tube  of  the 
solution  22  cm.  long  gives  a  rotation  of  io°? 

9.  10  cm.  of  a  certain  sugar  solution  gave  40  Ventske  units  in  a 
saccharimeter.     Calculate  the  concentration. 

10.  What  is  the  rotation  in  a  tube  of  carbon  bisulphide,  20  cm. 
long,  situated  in  and  parallel  to,  a  magnetic  field  of  strength  100? 

EXPERIMENT  XXVIII. 

Polarimetry,  Specific  Rotation. — Study  carefully  the  construction 
and  operation  of  the  polarimeter  (§§230-233). 

Prepare  the  assigned  solution  by  dissolving  a  carefully  weighed 
amount  of  the  active  solute  in  a  graduated  flask  which  is  filled  with 
distilled  water  to  TOO  c.c.  Fill  one  of  the  tubes  with  distilled  water 
and  make  several  readings  of  both  zero  positions,  with  both  verniers 
(if  the  instrument  has  two).  Pour  the  assigned  solution  into  a 
carefully  cleaned  tube  and  make  an  equal  number  of  readings  of  the 
position  of  the  analyzer. 

If  the  instrument  is  not  provided  with  a  thermostat  device,  great 
care  is  necessary  that  the  temperature  does  not  change  appreciably. 
Determine  similarly  the  rotation  of  several  other  concentrations 
of  the  assigned  substance.  From  the  mean  rotations,  calculate  the 
specific  rotations  and  plot  them  against  concentrations. 

If  time  permits  and  a  Lippish  instrument  is  used,  determine  the 
specific  rotation  for  several  other  wave  lengths  (§229). 

QUESTIONS. 

1.  Calculate   the   rotation   produced   by   a   meter   column   of  the 
first  solution. 

2.  What    is    the    purpose    of    (a)    both    zero    readings?      (6)    two 
verniers  ? 

3.  What  would  be  the  effect  of  using  white  light? 

4.  Construct    an    equation    from    your    observations    which    will 
represent  the  specific  rotation  for  different  concentrations. 


la^ 

I 


CHAPTER  VI. 
CHEMICAL  KINETICS.1 

234.  The  Law  of  Mass  Action. — With  certain  limitations 
discussed  later,  the  rapidity  of  chemical  or  physical  changes 
is  proportional  to  the  amount  changing.     The  validity  of  this 
law  in  many  systems  is  axiomatic,   for  we  may  imagine  the 
system  undergoing  change  to-  be  divided  into  equal  portions, 

d  the  total  change  per  unit  time  is  obviously  proportional 
to  the  number  of  parts.  But  if  we  mean  by  amount  the 
amount  per  unit  volume,  that  is,  the  concentration,  which  is 
the  usual  understanding  in  chemical  problems,  the  validity 
of  the  law  is  not  self-evident.  It  has,  however,  been  demon- 
strated by  experiments  mentioned  below,  and  we  will  accept 
it  as  the  foundation  of  this  chapter.  In  a  crude  fashion 
we  can  explain  the  law  if  we  consider  such  changes  from  a 
mechanical  point  of  view.  We  can  regard  the  changes  as  due 
to  the  impact  of  particles  or  their  proximity,  and,  therefore, 
the  greater  the  concentration,  the  greater  the  rapidity  of 
change. 

235.  A   chemical   or   physical   system   of  bodies   may   be 
regarded,  or  studied,  from  either  of  two  points  of  view.     One 
is  the  integral,  where  we  study  the  total  mass,  m;  volume,  v,  and 

ntropy  T?  (and  sometimes  energy  and  heat) ;  the  other  is  the 
differential,  where  a  small  portion  of  the  system  is  considered. 
We  then  consider  the  so-called  intensity  factors  which  are  the 
e  for  any  part  of  the  system  as  for  the  whole.  The  inten- 
sity factors  are  pressure,  p\  temperature,  t,  and  concentration, 
c.  In  place  of  concentration,  we  may  use  Gibbs'  potential  u, 
e  potential  of  any  system  being  the  work  required  to  intro- 
uce  unit  mass  (compare  electrical  potential  =work  required 

1  General  references  for  Chapters  VI  and  VII:  Meyer,  "Chemische  Reaktions- 
ischwindigkeit ";  Van't  Hoff,  Lectures,  vol.  i;  Mellor,  "Chemical  Statics  and 
Jynamics";  Chap.  I-VI;  Jiiptner,  vol  ii;  Walker,  Chap.  XXIV;  Nernst,  iii, 
:hap.  I,  V. 

s  225 


226  CHEMICAL    KINETICS. 

to  bring  up  unit  charge).1  Notice  that  the  intensity  factors 
are  the  derivatives  of  the  energy  with  respect  to  the  three 
magnitudes,  volume,  mass,  and  entropy. 

dW  dW  dW 

—r-  =  p;  —j—  =  t;  -j —  =  a  (see  Chap.  II)         (148) 
dv  drj  dm 

If  the  change  is  accompanied  by  alteration  of  the  intensity 
factors,  pressure  or  temperature,  or  alteration  of  the  potential 
from  without  the  system,  the  above  simple  law  may  obviously 
fail,  since  it  considers  only  the  integral  factors.  (Of  course 
the  potential  does  not  remain  constant,  for  the  change  is  due  to 
inequalities  of  potential  and  continues  until  the  potentials 
are  the  same  throughout  the  system.) 

236.  In  chemical  changes,  the  rapidity  of  change  is  deter- 
mined by  the  mass  per  unit  volume,  i.e.,  the  concentration. 
If  all  of  the  substance  does  not  take  part  in  the  reaction, 
that  portion  which  does  is  called  the  active  mass.2 

Suppose  we  have  a  chemical  system  where  the  temperature, 
total  pressure,  and  total  mass  remain  constant,  so  that  the 
above  law  holds.  We  will  suppose  a  reaction  taking  place 
which  we  will  represent  schematically  as 

HiA  +  HzB+nf-       —  n/A' +  n2'B' +  n3'C'  -    -(149) 

Where  A,  B,  C,  ---  A',  B't  C',  -  -  are  the  reacting  molecules 
and  nlf  na,  w3,  —  etc.,  are  the  numbers  of  these  molecules. 

The  rapidity  with  which  the  left-hand  side  changes  will  be 
proportional  to  the  number  of  each  of  the  reacting  molecules, 
and  therefore  to  their  product,  or,  c™  Xc2n2  Xc3M3 —  if  the 
concentration  of  A  is  clt  of  B  is  ca,  etc.,  for  ntA  is  equivalent 
to  A .  +  A  —  n  times.  Hence  in  the  total  product  we  have  cxni 

If  the  above  are  the  initial  concentrations,  and  after  time,  /, 
the  change  in  the  concentrations  is  x,  the  velocity  at  this  time 
will  be 

dx 
u  =  k(cl-x}^(c2-x}n*---  =  -^  (150) 

1  Phil.   Mag.,    Nov.,    1908,    p.  818;    Jiiptner,    ii,  2,  pp.    244,    245.     Nernst, 
pp.  612,  614. 

2  Walker,  p.  257. 


MONOMOLECULAR    REACTIONS.  227 

where  k  is  a  constant  called  tbe  velocity  constant.  The  principle 
described  above  and  expressed  in  this  equation  is  known  as  the 
law  of  mass*action  and  was  first  formulated  by  Guldberg  and 
Waage.1  This  differential  equation  can  be  integrated  in  a 
few  simple  cases. 

237.  (A]  Monomolecular  Reactions. — The  reaction  con- 
sists of  the  decomposition  of  a  single  molecule;  that  is,  it  is 
monomolecular . 

dx 

.-.  u  =  -     =  k(c-x)  (151) 


(i53) 

We  shall  consider  for  the  present  complete  reactions;  that 
is,  reactions  where  the  final  concentration  of  the  original 
substance  is  practically  zero,  c,  therefore,  is  both  the  initial 
concentration,  and  the  total  change  in  concentration. 

c—  #!  is  the  concentration  which  is  still  to  change  at  time  /x 
and  c—x2  is  the  unchanged  concentration  at  time  t2.  In  the 
experiments  which  follow,  we  do  not  measure  the  concentra- 
tions directly,  but  certain  physical  magnitudes  proportional 
to  the  concentrations.  If  we  plot  these  observations  against 
the  time,  we  can  find  the  above  ratios  without  working  out 
the  actual  concentrations.  It  is  customary  to  use  the  minute 
as  the  unit  of  time  in  chemical  mechanics. 

If  we  can  begin  our  observations  at  the  beginning  of  the 
reaction,  we  can  set  tl  =  o,  xl  =o 

•••*-'-**  ('545 


and  we  only  need  the  ratio  of  the  total  change,  to  the  change 
remaining  at  time  /.     Notice  that  the  equation  for  the  velocity 
1  Ostwald's  "  Klassiker,"  104. 


228  CHEMICAL    KINETICS. 

constant  of  a  monomolecular  reaction  does  not  involve  the 
actual  concentration. 

If  we  know  k  for  a  particular  reaction,  we  can  find  the 
velocity  of  the  reaction  (change  in  concentration  per  minute) 
by  multiplying  the  instantaneous  concentration  by  k  (Equa- 
tion 151). 

238.  Period  or  Time  Constant.  —  Let  T  be  the  time  required 
for  the  reaction  to  be  half  completed 


T  is  often  called  the  time  constant  or  period  of  the  reaction. 

239.  Exponential  Formula.  —  We  may  write  Equation  i  54 

c  —  x 


-kt=\n 
•    e-kt. 


c 
c  —  x 


c 

.  .  c  —  x  =  ce-kt  and  x  =  c(i  —  e-ki)  (156) 

There  are  comparatively  few  reactions  whose  velocity  is 
neither  too  great  nor  too  small  for  laboratory  measurement  and 
this  is  particularly  true  of  monomolecular  reactions.  If  the 
resources  of  an  organic  laboratory  are  available,  a  very  good 
illustration  is  the  change  of  acetochloranilide  to  the  para-form.1 
The  various  radioactive  transformations  are  monomolecular 
(§§3  49>  3  51),  but  these  are  complicated  by  successive  reactions. 
The  hydrolysis  of  an  ester,  the  decomposition  of  benzol- 
diazonium  chloride  and  the  inversion  of  cane-sugar  behave 
as  monomolecular  reactions.  These  are  strictly  bimolecular 
reactions,  but  the  other  component  (water)  has  usually  so 
high  a  concentration  compared  with  the  other  substance  that 
it  may  be  regarded  as  constant  or  taking  no  part  in  the 
reaction. 

PROBLEMS  XIII. 

i.  In  a  certain  monomolecular  reaction,  the  initial  concentration 
is  5.  The  concentration  at  10  A.M.  is  three,  and  at  4  P.M.  is  one. 
Calculate  (a)  the  velocity  constant;  (b)  the  initial  velocity  of  the 
reaction ;  (c)  the  period ;  (d)  when  did  the  reaction  commence  ? 

1  Blanksma,  Rec.  trav.  Pays-Bas.,  1902,  xxi,  p.  366;  1903,  xxii,  p.  290. 


THE    CATALYSIS    OF    AN    ESTER.  22Q 

2.  The  period  of   a  certain  monomolecular  reaction  is  one    hour 
and   the  initial  concentration  is   10.     Find   (a)    concentration  after 
30  minutes;  (6)  concentration  after  two  hours;  (c)  velocity  of  the 
reaction  at  this  time ;  (d)  plot  a  curve  which  will  represent  the  progress 
of  this  reaction. 

3.  Derive  an  expression  connecting  T',  the  time  when  the  fraction 
i  /e  remains  unchanged,  and  k.     e  is  the  base  of  natural  logarithms. 
(Answer:  T'  =  i/k.) 

240.  The  Catalysis  of  an  Ester.1  Comparison  of  Strengths 
of  Acids.2 — If  an  ester,  such  as  methyl  acetate,  is  dissolved 
in  water,  a  large  portion  of  the  salt  ultimately  breaks  up  into 
acid  and  alcohol,  according  to  the  equation 

CH3OOC2H3  +  H2O  —  CH3COOH  +  CH3OH 

The  change  is  greatly  accelerated  if  an  acid  is  present, 
and  the  so-called  strongest  acids,  such  as  hydrochloric,  have 
the  greatest  accelerating  influence.  In  fact,  if  the  different 
acids  are  tabulated  in  the  order  of  their  various  properties, 
such  as  replacing  power,  dissociation,  inversion  of  sugar,  etc., 
this  characteristic  is  in  the  same  order.  All  of  these  prop- 
erties are  therefore  said  to  be  a  measure  of  the  strength  or 
affinity  of  the  acid. 

We  shall  see  later  that  the  characteristic  of  an  acid  solution 
is  the  presence  of  free  hydrogen  ions,  or  atoms  with  positive 
charges,  and  that  the  strength  of  the  acid  is  a  measure  of  the 
number  of  hydrogen  ions.  The  acid  apparently  takes  no  part 
in  the  reaction ,  but  merely  accelerates  it .  Such  an  accelerating 
agent  is  called  a  catalyser. 

EXPERIMENT  XXIX. 
Catalysis  of  an  Ester. 

We  shall  follow  the  above  reaction  by  measuring  the  acid  formed 

.at   different   times,    and   from  the  observation  we  shall  calculate  the 

velocity  constant.     Prepare  the  following  apparatus.     A  thermostata 

(§81)   at   about   25°',   containing  a   75   c.c.  Erlenmeyer  flask  with  a 

paraffined  cork,  and  a  small  bottle  of  methyl  acetate.      Both  should 

1  Ostwald,  Journ,  prakt.  Chem.,  1883,  xxviii,  p.  449;  Hemptinne,  Zeit. 
phys.  Chem.,  1899,  xxxi,  p.  35. 

*  Walker,  Chap.  XXV. 

3  A  thermostat  is  recommended  in  this  and  following  experiments,  but  a  simple 
water-bath  whose  temperature  is  maintained  within  .5°  gives  fair  results,  and 
even  this  may  be  dispensed  with  if  the  temperature  of  the  room  is  very  constant 
and  the  liquids  are  not  subjected  to  appreciable  changes  of  temperature. 


230  CHEMICAL    KINETICS. 

be  submerged  to  their  necks  by  weights  or  frames.  Two  100  c.c. 
Erlenmeyer  flasks,  three  2  c.c.  pipettes.  A  burette  and  solution  of 
approximately  n/2o  barium  hydroxide  ( §90),  whose  strength  has  been 
carefully  determined  by  titration  against  a  standard  hydrochloric 
or  other  acid  solution.  An  abundance  of  CO2  free  water  (§90).  All 
the  vessels  should  be  scrupulously  clean. 

Prepare  a  semi-normal  solution  of  the  assigned1  acid  and  put  50  c.c. 
in  the  small  flask.  In  each  of  the  larger  flasks  pour  about  30  c.c. 
of  CO2-free  water. 

When  all  is  ready,  fill  a  2  c.c.  pipette  with  the  methyl  acetate  and 
discharge  it  into  the  acid  solution.  Mix  thoroughly,  and  imme- 
diately withdraw  2  c.c.  in  one  of  the  other  pipettes  and  discharge  into 
the  water  in  one  of  the  larger  flasks.  Note  the  exact  time  when  the 
reaction  is  stopped  by  this  dilution.  Immediately  titrate  with  the 
baryta  solution,  using  phenolphthalein  as  an  indicator.  Ten  minutes 
later  withdraw  2  c.c.  more,  and  repeat  the  dilution  with  observation 
of  time  and  titration.  So  continue,  increasing  the  length  of  time 
between  the  withdrawal  of  samples  as  the  rate  of  change  decreases. 
The  final  titration  should  be  at  least  two  days  after  the  first. 

Plot  your  results  with  times  as  abscissae  and  titrations  as  ordinates, 
and  draw  a  smooth  curve. 

Since  only  the  ratio  of  concentrations  is  required,  titrations  may 
be  substituted.  The  total  change  in  concentration,  i.e.,  the  initial 
concentration,  c,  will  be  equal  to  the  total  acetic  acid  produced  which 
is  measured  by  the  total  change  in  titration  T^  —  T0.  T0  is  the  initial 
titre  in  c.c.  and  T^  the  final.  The  concentration  of  the  unchanged 
ester,  c  —  x,  at  any  time  /  will  be  proportional  to  the  remaining  change 
in  titration,  T^—Tt  where  Tt  is  the  titre  at  that  instant.  .'.,  by 
Equation  i  54 

,          2  . "?  i  •*    00  —  *  O  /  .\ 

£  =  ^logr         Tt  (i54') 

t          1  GO  —  1 1 

If  there  is  any  uncertainty  about  T0,  the  titre  before  any  acetic 
acid  has  formed,  find  the  titre  of  50  c.c.  of  the  assigned  half-normal 
acid  plus  2  c.c.  of  water.  Or,  Equation  i  53  may  be  used. 

Choose  5  typical  ordinates  of  the  smooth  curve,  find  the  correspond- 
ing times  and  calculate  the  values  of  k.  Take  the  mean.2 

QUESTIONS. 

1.  What  was  the  initial  velocity  of  the  reaction?     In  what  units 
is  it  expressed? 

2.  Calculate  from  the  mean  velocity  constant  the  time  when  the 
reaction  was  one-half  completed  and  compare  with  the  experimental 
result  obtained  from  the  curve.      (Time  constant.) 

3.  Calculate  the   final  concentration  (gram  equivalents  per  litre) 
of  water,  acetic  acid,  and  alcohol. 

1  This  experiment  may  be  varied  by  assigning  different  acids  to  successive 
students.     When  all  have  completed  the  experiment,  the  different  values  of  the 
velocity  constant  may  be  given  the  entire  class  for  comparison. 

2  In  almost  all  the  experiments  of  this  chapter  a  so-called  constant  is  determined 
which  by  the  theory  should  be  invariable.     The  experiment  may  give  values 
varying  10%  or  more,  but  such  lack  of  agreement  should  be  compared  with  the 
great  variation  in  the  individual  factors  from  which  it  is  calculated  and  their 
possible  errors  (see    §§,  16-20). 


INVERSION    OF    CANE-SUGAR.  231 

241.  The  Inversion  of  Cane-sugar.1  Comparison  of  the 
Strengths  of  Acids.2 — Cane-sugar,  dissolved  in  acidulated 
water,  gradually  breaks  up  into  a  mixture  of  glucose  and 
fructose  (dextrose  and  laevulose) ,  which  is  called  invert  sugar, 
the  reaction  being 

CI2H220II+H2O  =  2C6HI2O6 

The  cane-sugar  on  the  left-hand  side  gives  a  right-handed 
rotation  of  the  plane  of  polarization  of  polarized  light  (§225) 
and  the  mixture  of  the  two  isomers  on  the  right  gives  a  net 
left-handed  rotation.  Therefore,  the  progress  of  the  reaction 
can  be  studied  by  observing  the  rotation. 

The  acid  apparently  takes  no  part  in  the  reaction,  but 
acts  as  an  accelerating  or  so-called  catalytic  agent.  A  compari- 
son of  the  rates  of  inversion  produced  by  different  acids,  shows 
that  this  property  varies  with  the  strength,  or  affinity,  of  the 
acid,  which  we  will  find  later  varies  with  the  concentration  of 
hydrogen  ions  (see  preceding  experiment). 

The  rotations  produced  by  both  cane  and  invert  sugars  are 
closely  proportional  to  the  concentrations.  Since,  therefore, 
in  the  formula  for  the  velocity  constant  we  have  only  the 
ratio  of  concentrations,  we  may  substitute  total  change  in 
rotation  $  for  total  change  in  concentration  (or  initial  con- 
centration) ,  c,  and  change  in  rotation  ^  at  any  time  t,  for  the 
corresponding  change  in  concentration  x.  Therefore,  if  we 
count  time  from  the  instant  that  we  begin  to  observe  the 
rotation,  by  Equation  154 


k,  the  velocity  constant,  is  the  factor  by  which  the  concentra- 
tion of  unchanged  sugar  must  be  multiplied  to  obtain  the 
instantaneous  velocity  of  inversion. 

1  Wilhelmy,  Pogg.  Ann.,  1850,  Ixxxi,  p.  413;  p.  499. 
*  Walker,  Chap.  XXV. 


232  CHEMICAL    KINETICS. 

EXPERIMENT  XXX. 
Inversion  of  Cane-sugar.1 

We  will  use  a  comparatively  dilute  solution  so  that  the  change  in 
concentration  of  the  water  may  be  inappreciable,  and  the  reaction 
may  behave  as  monomolecular. 

See  §§230-233  for  a  full  description  of  the  construction  and 
operation  of  the  polarimeter.  The  rotation  changes  considerably 
with  the  temperature  and  hence  the  rotation  tube  should  properly 
be  surrounded  by  a  thermostat  jacket,  but  this  is  not  absolutely 
necessary  if  the  room  temperature  is  maintained  very  constant. 
Mix  thoroughly  equal  volumes  of  the  assigned  sugar  solution  (e.g., 
20%  =  20  gr.  in  100  c.c.),  and  the  assigned  half-normal  acid  solution. 
Fill  a  polarimeter  tube  as  quickly  as  possible,  and  determine  the 
rotation.  If  the  instrument  has  a  double  vernier,  there  will  only  be 
time  to  read  one.  After  a  few  minutes,  read  the  rotation  again,  and 
so  continue,  observing  carefully  both  rotations  and  time,  and  making 
the  intervals  longer  as  the  changes  become  less.  The  later  readings 
should  be  at  intervals  of  several  hours  and  the  last  after  two  days. 

Plot  rotations  against  times  as  abscissae.  The  later  rotations  will 
of  course  be  negative.  Draw  a  smooth  curve  through  the  obser- 
vations, choose  five  points  on  the  curve  and  for  each  calculate  the 
velocity  constant  and  take  the  mean. 

In  the  preceding  experiment  we  showed  how  the  reaction  could 
be  followed  from  the  beginning.  We  could  do  the  same  b,ere  by 
finding  the  rotation  for  a  solution  where  the  acid  is  replaced  by  water. 
It  is  simpler,  however,  to  consider  the  time  of  the  first  observation 
as  the  beginning  of  the  reaction.  The  initial  concentration  is  then 
of  course  different  from  the  original  solution,  but  this  is  not  material^ 

QUESTIONS. 

1.  How  many  grams  of  cane-sugar  would  be  inverted  per  minute 
in  a  litre  of  10%  solution,  if  the  acidification  was  the  same  as  in  this 
experiment  ? 

2.  Compare  the  experimental  and  calculated  values  of  the  time 
constant  of  this  reaction. 

3.  Show  that  no  error  is  introduced  by  considering  the  change  in 
rotation  proportional  to  the  change  in  concentration,  although  the 
specific  rotation  (§226)  of  the  mixture  constituting  invert  sugar  is 
less  than  that  of  cane-sugar. 

242.  Hydrolysis  of  a  Salt.2 — A  salt  composed  of  a  weak 
base  and  a  strong  acid,  such  as  ferric  chloride,  copper  sulphate 
or  carbamide  hydrochloride,  partially  breaks  up  when  dis- 
solved, into  the  acid  and  base;  for  example, 

FeCl3  +  3  H2O  =  Fe(OH) 3  +  3  HC1 

1  The  experiment  may  be  varied  by  assigning  different  half  normal  acids  (and 
the  same  sugar  solution)  to  successive  students.     When  all  have  completed  the 
experiment  the   different   velocity   constants  should   be   given  the   class    for 
comparison. 

2  See  also  Experiments  XXXI  and  XXXIV;    and  §§245,  261;  Mellor,  pp. 
207-216;  Juptner,  ii,  pp.  79-83;  Walker,  Chap.  XXVII. 


HYDROLYSIS    OF    A    SALT. 


233 


This  phenomenon  is  known  as  hydrolysis  or  hydrolytic  disso- 
ciation. The  acid  is  much  the  stronger  and  the  solution  shows 
an  acid  reaction.  Therefore,  it  should  produce  inversion  of 
sugar  and  catalysis  of  an  ester,  and  such  will  be  found  to  be 
the  case  if  the  hydrolysis  is  sufficient.  The  degree  of  hydroly- 
sis is  measured  by  the  ratio  of  the  acidity  of  the  solution,  to 
the  strength  of  the  acid,  if  the  hydrolysis  was  complete.  It 
must  therefore  equal  the  ratio  of  the  velocity  of  inversion,  or 
catalysis,  in  a  solution  of  the  salt,  to  the  velocity  in  a  solution 
of  the  resulting  acid  of  strength  equivalent  to  that  of  the 
salt.  The  ratio  of  the  velocities  will  evidently  be  the  ratio 
of  the  velocity  constants.  Little  acid  is  formed  unless  the 
base  is  very  weak,  and  therefore  in  this  case  only  is  the  method 
accurate. 

PROBLEMS  XIV. 

1.  Arseniuretted    hydrogen   decomposes,    monomolecularly,    at    a 
moderately   high   temperature.      The    velocity   constant   at    367°   is 
.0034.      (a)    How  long  a  time  will  be  required  for  6  grams  to  be  de- 
composed out  of  an  initial  mass  of  10  grams?      (6)   At  what  rate  is  it 
changing  at  the  end  of  this  time  ?      (c)   Plot  a  curve  which  will  show 
the  progress  of  the  reaction. 

2.  The  velocity  constant  of  a  20%  sugar  solution  in  n/2  lactic  acid 
at  25°  is  .000023.     Find  (a)  the  concentration  one  day  after  mixing ; 
(6)  the  velocity  of  inversion  at  this  time ;  (c)  the  time  required  for  in- 
version of  half  the  sugar,     (d)   If  the  velocity  constant  of   n/2  sul- 
phuric acid  is  .005,  compare  the  strengths  of  the  two  acids. 

3.  A  certain  sugar  solution  gave  the  following  rotations,  t  minutes 
after  an  acid  was  added. 

t  angle 

o  46.75 

30  41.00 

90  3°-75 

I2O  26.OO 

GO  -18.70 

4.  The   velocity  constant  of  normal  hydrochloric  acid  at   25°  is 
.015,  that  of  n/4  urea  hydrochloride  is  .0006.      What  is  (a)  the  degree 
of  hydrolysis  of  the  latter?      (b)  the  concentration  of  free  acid? 

5.  The  activity  (in  arbitrary  units)  of  a  certain  volume  of  radium 
emanation  was  as  follows: 


Calculate  the  velocity 
constant. 


Time  (hrs.) 
o 

20.8 

187.6 

354-9 
521.9 
786.9 


Activity. 
100 

85.7 

24. 

6.9 

!-5 

.19 


Calculate  the  velocity 
constant  (=  radio-ac- 
tive constant)  and  the 
period  (Meyer,  p.  25). 


234  CHEMICAL    KINETICS. 

EXPERIMENT  XXXI. 
Hydrolysis  of  Carbamide  Hydrochloride.1 

In  100  c.c.  of  half-normal  hydrochloric  acid  solution,  dissolve 
an  equivalent  amount  of  carbamide.  Repeat  Experiment  XXIX 
(catalysis  of  methyl  acetate),  with  this  solution  substituted  for  the 
acid.  Carry  on  a  parallel  experiment  with  half-normal  hydro- 
chloric acid  solution. 

From  the  ratio'  of  the  two  average  velocity  constants,  determine 
the  degree  of  hydrolysis. 

The  degree  of  hydrolysis  may  also  be  determined  by  Experiment 
XXX  (velocity  of  inversion  of  cane-sugar),  the  acid  solution  being 
replaced  by  an  equivalent  solution  of  carbamide  hydrochloride. 

QUESTIONS. 

1.  It  is  shown  in  more  complete  treatises  that  the  presence  of  a 
neutral  salt  retards  the  reaction.      How  could  you  experimentally 
correct  for  the  effect  of  the  neutral  salt  in  the  above  experiment  ? 

2.  Explain  which  method  you  consider  preferable. 

EXPERIMENT  XXXII. 
The  Decomposition  of  Diazonium  Salt.2 

In  the  previous  experiment  we  have  followed  the  reaction  by 
chemical  analysis  or  measurement  of  the  rotatory  polarization.  In 
this  experiment  we  shall  use  a  third  method,  namely,  measurement 
of  the  volume  of  gas  produced. 

Benzol  diazonium  chloride,  C6H5N2C1,  is  prepared  at  zero  degrees, 
in  aqueous  solution.  Upon  raising  the  temperature  of  the  solution, 
the  salt  reacts  with  water,  forming  phenol,  hydrochloric  acid,  and 
nitrogen  according  to  the  equation: 


C6H5N2C1  +  H20=C6H5 

We  shall  use  so  large  an  excess  of  water  that  the  reaction  is  prac- 
tically monomolecular. 

The  amount  of  decomposition  is  determined  from  the  volume  of 
nitrogen  liberated.  V^  =  final  volume  of  nitrogen,  Vu  =  volume  at  /: 
minutes,  Vt2  =  volume  at  /2  minutes.  By  Equation  i  53 


k  is  the  velocity  constant  or,  the  velocity  of  the  reaction  at  any 
instant  is  k  times  the  instantaneous  concentration  of  unchanged 
salt. 

A  100  c.c.  Bunsen  gas  washing  bottle,  with  a  long  wide  neck, 
should  be  provided  with  a  Witt  stirrer  (§82)  which  passes  through  a 
mercury  gas  seal  (§85).  The  side  tube  should  be  connected  to  a  gas 
burette  (§83).  A  thermostat  (§81)  at  about  30°  should  be  filled 
with  water  to  such  a  height  that  this  bottle  may  be  immersed  to  the 

1  Walker  and  Bredig,  Zeit.  phys.  Chem.,  1889,  iv,  p.  319;  1894,  xiii,  p.  214; 
1900,  xxxii,  p.  348. 

2  Findlay,  "Practical  Physical  Chemistry,"  p.  237. 


BIMOLECULAR    REACTIONS.  235 

neck.  A  small  electric  motor  should  be  adjusted  to  drive  the 
stirrer  by  a  thread  belt.  An  ice-water  bath  should  also  be  ready. 
Having  prepared  the  bottle,  place  it  in  the  ice-bath,  and  also  two 
100  c.c.  Erlenmeyer  flasks  and  a  litre  graduated  flask  containing 
goo  c.c.  of  water.  Prepare  the  benzol  diazonium  chloride  solution 
as  follows:  In  one  flask  dissolve  6.64  gr.  of  aniline  in  21.4  c.c.  of 
hydrochloric  acid  (specific  gravity  1.16).  In  the  other  dissolve 
4.9  gr.  of  sodium  nitrite  in  75  c.c.  of  water.  Pour  the  latter  solution 
into  the  former  very  slowly,  and,  finally,  pour  the  resulting  solution 
into  the  large  flask. 

Remove  the  stopper  and  stirrer  of  the  bottle  and  pour  in  about 
50  c.c.  of  this  diazonium  salt  solution.  Replace  the  stopper  and 
stirrer,  making  sure  that  the  mercury  seal  is  tight,  and  transfer  the 
whole  to  the  thermostat.  Do  not  connect  the  side  tube  to  the  gas 
burette  until  a  few  minutes  have  elapsed  and  the  solution  has 
acquired  the  temperature  of  the  thermostat.  Then  connect  to  the 
gas  burette,  arrange  the  thread  belt  and  motor  so  that  stirring  is 
uniform,  and  make  an  initial  reading,  Vti,  of  the  gas  burette,  and 
notice  the  time.  Read  the  gas  burette  and  observe  the  time  at 
intervals,  at  first  short,  and  later  longer  as  the  rate  of  change  de- 
creases. When  the  readings  have  become  almost  constant,  transfer 
the  gas  bottle,  still  connected  to  the  gas  burette,  to  hot  water, 
where  the  reaction  will  soon  become  complete.  Return  the  bottle 
to  the  thermostat  and  record  the  reading  of  the  gas  burette.  This 
will  be  FQC  .  Observe  the  temperature  of  the  gas  burette  and  the 
barometer.  Plot  the  volumes  against  the  time,  in  minutes,  as 
abscissa?.  Draw  a  smooth  curve  through  the  observations.  Choose 
five  well-distributed  ordinates  (volumes),  find  the  corresponding 
times,  and  taking  them  by  pairs,  calculate  k.  Find  the  mean  value 
of  k. 


QUESTIONS. 

1.  Explain  whether  it  would  be  better  to  reduce  the  volumes  to 
standard  conditions  before  substitution  in  the  formula. 

2.  What  would  be  the  initial  change,   in  grams  per  minute,   in 
a  fresh  litre  solution  of  10  grams  of  this  salt  at  this  temperature? 

3.  Compare  the  experimental  and  calculated  values  for  the  time 
when  the  reaction  was  half  completed  (time  constant). 

243.  Bimolecular  Reactions. — The  next  simplest  case  to  a 
monomolecular  reaction  is  one  where  two  molecules  are 
involved,  and  such  is  called  a  bimolecular  reaction.  This  class 
of  reactions  may  further  be  subdivided  into  (A)  those  where 
the  initial  concentrations  of  both  molecules  are  the  same  and 
(B)  those  where  the  initial  concentrations  are  different. 

(A)   In  Equation  (150)  let  cz  =  c2,  =  c,  nl  =  n2  =  i 

•'•  u=-    =  k(c-x)2  (157) 


236 

Integrating 


CHEMICAL    KINETICS. 


t,        2 
'  f *  -      r^v 

J     I  «-)* 

ft        I/ 


If  we  can  follow  the  reaction  from  the  beginning,  c,  the 
initial  concentration  is  equal  to  the  total  change  in  concen- 
tration, for  we  are  still  considering  reactions  which  are  practi- 
cally complete.  If  we  then  count  our  time  from  the  commence- 
ment of  the  reaction,  and  set  xl  and  tl  equal  to  zero,  we  have 

x 

*=te*F^  (I59) 

xf  (c  —x}  is  merely  a  ratio  and  can  be  expressed  in  any  conven- 
ient units,     x  is  proportional  to  the  change  previous  to  the 
time  t  and  c  —  x  to  the  change  after  this  time. 
(B}.  Different  initial  concentrations,  c1  and  c2. 

For  simplicity,  we  will  commence  to  count  time  at  the  beginning  of 
the  reaction.  x 


.(kit-    \      -g- 

J  I    (cl-x}(c2-x) 

o  <J 


Separating  into  partial  fractions, 


kt 


it-  ;*> 

/dx  I      dx 

,-,'J  „- 


\J  \S 

i    r,  cI-x~\x      2.3   , 

=  —       -    In  —  = ^—  log 

cl-c2\_     c2-x\0      cl-c2 


.'.  k  = 


gf3- 

•i-O* 


log 


C2  (CI  -  X) 
Ci  (C*  -  X) 


(161) 


SAPONIFICATION    OF    ESTERS. 


237 


:,  is  here  the  greater  initial  concentration.  The  action  pro- 
jeds  until  the  weaker  substance  is  practically  exhausted ;  that 
;,  until  x,  the  change  in  concentration,  is  practically  equal 
c2,  the  smaller  initial  concentration. 

Most  bimolecular  reactions  proceed  so  rapidly  that  their 
mrse  cannot  be  followed,  but  there  are  a  few  reactions  of 
>roper  velocity,  some  of  which  are  discussed  in  the  pages 
following. 

244.  Saponification  of  Esters.1  Strength  of  Bases.2 — When 
solution  of  an   ester  such  as  ethyl  acetate  is  mixed  with 
solution   of   a  strong  base,  practically  all  of  the  ester  is 
dtimately  broken  up  into  alcohol  and  the  acetate  corres- 
>onding  to  the  base.     This  phenomena  is  called  saponifica- 
m.     The  rapidity  of  the  saponification  depends  upon  the  base 
ised  and  if  the  accelerating  powers  of  various  bases  are  deter- 
lined,  comparative  values  are  obtained  similar  to  those  given 
)y  the  replacing  power,  electrical  conductivity,  etc.     There- 
fore, the  accelerating  power,  that  is,  the  velocity  constant, 
is  said  to  be  the  measure  of  the  strength  of  a  base.     Just  as 
he  strengths  of  acids  is  proportional  to  the  concentration  of 
lydrogen  ions,  so  the  strength  of  bases  is  proportional  to  the 
mcentration  of  the  radical  OH',  which,  as  we  shall  see  later, 
exists  in  solutions,  with  a  charge  of  negative  electricity.     Since 
;he  OH'  ions  have  a  marked  effect  upon  the  velocity  of  the 
^action,  they  are  said  to  act  as  pseudo-catalysers,  the  prefix 
;ing  added  because,  unlike  true  catalysers,  they  chemically 
ike  part  in  the  reaction. 

EXPERIMENT  XXXIII. 
Velocity  of  Saponification  of  Ethyl  Acetate.3 

We  will  watch  the  progress  of  the  reaction  by  the  disappearance 
)f  the  alkali,  and  from  our  observations  we  will  calculate  the  velocity 
mstant  k;  that  is,  the  factor  by  which  the  product  of  the  instanta- 
jous  concentrations  must  be  multiplied  to  obtain  the  instantaneous 

1  Warder,  Berichte,  1881,  xiv,  p.  1311. 

2  Walker,  Chap.  XXV. 

3  This  experiment  may  be  varied  by  assigning  different  strong  bases  to  suc- 
jssive  students.     When  all  have  completed  the  experiment,  the  class  may  be 

jiven  all  the  velocity  constants  obtained,  for  comparison  of  the  strengths  of  the 
lifferent  bases. 


238  CHEMICAL    KINETICS. 

velocity  of  reaction.  Let,  for  example,  n/40  sodium  hydroxide  be 
the  assigned  alkali  and  n/  50  ethyl  acetate  be  the  ester.  Clean  care- 
fully one  100  c.c.  and  three  75  c.c.  Erlenmeyer  flasks,  a  10  c.c.,  a  2*0 
c.c.,  and  a  50  c.c.  pipette,  and  two  burettes.  Prepare  about  500  c.c. 
of  approximately  n/2o  baryta  solution  which  should  be  standardized 
with  an  acid  solution  of  known  strength.  Prepare  also  500  c.c.  of 
n/5o  HC1  solution  and  put  20  c.c.  of  the  latter  into  two  of  the  small 
flasks.  Prepare  the  n/4o  sodium  hydroxide  and  n/$o  ethyl  acetate 
solutions.  Put  50  c.c.  of  the  latter  into  the  remaining  small  flask, 
cork  with  a  paraffined  stopper  and  place  in  a  thermostat  at  about 
25°.  Put  50  c.c.  of  the  sodium  hydroxide  solution  in  the  100  c.c. 
flask  and  set  this  also  in  the  thermostat. 

When  all  is  ready  and  the  thermostat  has  come  to  a  steady  tem- 
perature, pour  the  ester  into  the  alkali,  mix  thoroughly  and  note 
the  exact  time  of  mixing.  As  soon  as  possible,  withdraw  10  c.c.  of 
the  mixture  by  means  of  the  pipette  and  discharge  into  the  20  c.c.  of 
w/5o  acid,  where  the  reaction  will  cease  at  once.  Note  the  mean  time 
of  mixing  with  the  acid.  Five  minutes  later  withdraw  10  c.c. 
more  and  discharge  into  the  other  flask,  noting  the  time.  Titrate 
each  solution  against  the  baryta  solution,  using  phenolphthalein  as  an 
indicator.  Empty  and  clean  the  flasks  and  refill  them  with  20  c.c. 
of  the  w/5o  acid.  After  ten  minutes,  withdraw  another  10  c.c.  sample 
and  titrate;  then  another  sample  30  minutes  later;  another  after 
one  hour;  another  after  two  hours;  another,  if  possible,  after  six 
hours,  and,  finally,  one  after  twenty-four  hours. 

Plot  your  titrations  in  c.c.  of  baryta  solution  against  the  times  as 
abscissae,  and  draw  a  smooth  curve  through  the  points  obtained. 

The  initial  concentrations  cx  and  c2  are  known.  Since  we  have 
assumed  c2  to  be  the  smaller,  it  also  represents  the  total  change  in 
concentration,  hence  c2  —  x  for  any  time  t  can  be  found  from  the 
difference  between  the  titration  at  this  time  and  the  final  titration. 
Suppose  this  difference  is  N  c.c.  and  the  concentration  of  the  baryta 
solution  is  i/n  (approximately  «/2o) 

AT 


Substitute  in  Equation  161. 

It  is  also  possible  to  calculate  k  without  knowing  the  strength  of 
the  baryta  solution.  c2  is  proportional  to  the  total  change  of  titre, 
c2  —  x  is  proportional  to  the  change  in  titre  after  time  t.  For  d 
may  be  substituted  the  c2  titre  multiplied  by  the  ratio  d/c2  which 
is  known.  ct—  x  is  the  latter  minus  the  change  in  titre  before 
the  time  t.  These  four  titres,  or  equivalent  titres,  may  be  sub- 
stituted for  the  concentrations  in  the  logarithmic  expression.  For 
Ci  —  c2  the  true  value  must  be  substituted  (.0025,  if  the  above  con- 
centrations are  used). 

If  cl  and  c2  had  been  equal,  Equation  158  or  Equation  159  should 
have  been  used. 

From  the  ordinate  (titration)  and  abscissa  (time)  of  four  typical 
points  on  the  curve,  calculate  the  velocity  constant  and  take  the 
mean. 

In  Experiment  XLIX  the  velocity  of  saponification  is  determined 
by  an  electrical  method. 


HYDROLYSIS. 


239 


QUESTIONS. 

1.  Calculate  the   velocity  of  reaction  ten  minute1   after  it   com- 
menced (express  in  gram-equivalents,  per  litre,  per  minute). 

2.  Compare  the  experimental  and  calculated  values  for  the  time 
when  the  reaction  was  half  completed.      (Time  constant.) 

245.  Hydrolysis.  Strong  Base  and  Weak  Acid.1 — A  salt 
composed  of  a  strong  base  and  a  weak  acid  partly  decomposes 
in  aqueous  solution  into  the  original  base  and  acid  and  shows 
an  alkaline  reaction.  The  degree  of  hydrolysis  can  be  found 
by  measuring  the  velocity  of  saponification  when  the  salt 
solution  replaces  the  base  solution  of  the  last  experiment. 
The  ratio  of  the  velocity  constant  to  that  of  an  equivalent 
solution  of  the  pure  base  is  the  degree  of  hydrolysis,  for  the 
velocity  constant  of  saponification  is  a  measure  of  the  strength 
of  an  alkali  (§244). 

PROBLEMS  XV. 

i.  A  100  c.c.  mixture  of  ethyl  acetate  and  sodium  hydroxide 
solutions  gave  the  following  titrations  with  ^723.3  hydrochloric  acid. 


o. 
4.89 

IO-37 
28.18 

oo 


titre 
61.95 

50-59 
42.40 

29-35 
14.92 


Calculate  (a)  final  concentration  of  base ;  (6)  initial  concentration  of 
The  difference  will  evidently  be  the  initial  concentration  of 
jster.      (c)   Calculate  the  velocity  constant. 

2.  The  velocity  constant  of  methyl  acetate  and  barium  hydroxide 
at    9.4°  is   2.14.      What   is  (a)  the   initial   velocity  of  saponification 
if  normal  base  is  mixed  with  half-normal  ester?      (6)  What  are  the 
concentrations  after  10  minutes?      (c)    When  is  the  concentration  of 
ester  reduced  to  one-half? 

3.  .3%    of   normal    potassium     cyanide     solution   is   hydrolyzed. 
What  is  the  velocity  constant  of  this  solution  for  saponification,  if 

"ic  velocity  constant  of  normal  potassium  hydroxide  is  2.3? 

EXPERIMENT  XXXIV. 
Hydrolysis  of  Sodium  Carbonate. 

Sodium  carbonate  hydrolyzes  in  aqueous  solutions, 
Na2C03  +  2H20=2NaOH+H2CO3 

'repare    tenth-normal    solutions    of    sodium    carbonate    and    ethyl 
:ate  and  place  the  solutions  in  a  thermostat  at  about  2  5°.     Provide 

1  See  also  Experiment  XXXI  and  references  for  §  242. 


240  CHEMICAL    KINETICS. 

all  the  apparatus  used  in  the  previous  experiment  except  the  baryta 
solution.  The  reaction  is  so  much  slower  than  when  a  base  is  used 
that  it  is  not  necessary  to  check  the  reaction  by  adding  an  excess  of 
acid.  The  content  of  alkali  is  determined  by  direct  titration  with 
w/5o  or  w/ioo  hydrochloric  acid  solution. 

When  ready,  pour  50  c.c.  of  one  solution  into  the  other,  note  the 
time,  and  thoroughly  mix.  Immediately  pipette  5  c.c.  of  the 
mixture  into  one  X)f  the  small  dry  Erlenmeyer  flasks,  note  the  time 
and  dilute  with  about  20  c.c.  of  water.  Add  a  few  drops  of  methyl- 
orange  and  immediately  titrate  against  the  hydrochloric  acid 
solution. 

Ten  minutes  later  analyze  similarly  5  c.c.  more,  and  so  continue, 
at  longer  and.  longer  intervals,  until  the  readings  change  but  little. 
Make  the  final  analysis  after  two  days.  Plot  titrations,  against 
time  in  minutes.  Take  four  well-distributed  points  on  the  curve, 
and  for  each  calculate  the  velocity  constant  from  Equation  159. 

If  the  above  concentrations  are  used,  £=.05.  Draw  horizontal 
lines  parallel  to  the  time  axis  through  the  initial  and  final  points  on 
the  curve.  At  the  times  corresponding  to  each  of  the  points  chosen 
on  the  curve  erect  perpendiculars.  The  ratio,  x/c  —  x,  is  obviously 
the  ratio  of  the  distance  between  the  curve  and  the  upper  hori- 
zontal, to  the  distance  between  the  curve  and  the  lower  horizontal. 
Find  the  mean  value  of  k. 

If  time  permit,  repeat  Experiment  XXXIII  with  tenth-normal 
sodium  hydroxide  and  ethyl  acetate.  (Weaker  concentrations  are 
used  in  that  experiment  in  order  to  secure  a  slower  reaction  and  one 
which  therefore  can  more  easily  be  followed.) 

The  ratio  of  the  two  mean  velocity  constants  will  be  a  measure  of 
the  degree  of  hydrolysis.  If  there  is  insufficient  time  for  repeating 
Experiment  XXXIII,  an  approximate  value  for  the  velocity  constant 
of  the  base  may  be  obtained  by  multiplying  the  value  obtained 
in  the  previous  experiment  by  the  ratio  of  the  concentration  of  the 
salt  in  this  experiment  to  that  of  the  base  in  the  earlier  experiment. 

QUESTIONS. 

1.  Devise  an   auxiliary  experiment   by  which  you   could   correct 
for  the  influence  of  the  neutral  salt. 

2.  What  error  is  there  in  estimating  the  velocity  constant  of  a 
reaction  at  a  certain  concentration,  by  multiplying  the  value  obtained 
with  a  lower  concentration  by  the  ratio  of  the  two  concentrations? 

246.  Determination  of  the  Order  of  a  Reaction.1 — By  the 

order  of  a  reaction  is  understood  the  number  of  molecules 
taking  part  in  the  reaction.  For  example,  the  decomposition 
of  calcium  carbonate 

CaCO3=CaO+CO2 

is  of  the  first  order  or  monomolecular.  The  saponification  of 
an  ester 

C2H5OOC2H3  +  NaOH  =  NaOOC2H3  +  C2 

1  Van't  Hoff,  i,  pp.  193-197;  Mellor,  §.21. 


ORDER    OF    A    REACTION.  241 

is  of  the  second  order  or  bimolecular,  etc.  In  many  cases, 
however,  it  is  difficult  to  determine  the  actual  number  of 
reacting  molecules  by  inspection.  If  the  reaction  is  sufficiently 
slow  to  enable  one  to  follow  it,  several  methods  are  available 
for  determining  its  velocity. 

247.  (A)  For  several  different  times,  calculate  the  velocity 
constant,  using  successively  the  formulae  for  monomolecular 
reactions,  bimolecular  reactions,  etc.     If  reasonably  constant 
values  of  the  velocity  constant  are  given  by  one  of  the  formulae, 
the  reaction  may  be  assumed  to  be  of  that  order. 

248.  (B)   Suppose  that  the  original  concentrations  of  the 
reacting  substances  are  the  same,  c,  and  the  order  of  the  re- 
action is  n.     The  velocity  of  the  reaction  at  any  time  /  will  be 

u  =  -j-  =  k  (c  —  x)  n  (162) 

where  x  is  the  change  in  concentration  at  this  time. 

Transposing  and  integrating  between  the  limits,  o  and  t,  o  and  x 

x 


Let  n  =  2  Let  n  =  3 


If  n  =  i,  the  above  formula  fails  and  we  must  reintegrate, 
when  we  shall  obtain  as  before  (Equation  154) 


—  •  111 


C  —  X 

Let  x  be  some  fraction  of  the  initial  concentration,  for  ex- 
ample, 1/2. 

n=  i      t=-j-ln  2  (164) 

K 

16 


242  CHEMICAL    KINETICS. 


(164') 


etc. 

Therefore,  if  the  reaction  is  of  the  first  order,  the  time  re- 
quired for  changing  a  definite  fraction  of  the  original  concen- 
tration is  independent  of  the  initial  concentration.  If 
the  order  is  the  second,  the  time  will  vary  inversely  with  the 
initial  concentration.  If  the  reaction  is  of  the  third  order, 
the  time  will  be  inversely  proportional  to  the  square  of  the 
initial  concentration,  and,  in  general,  it  can  easily  be  shown 
that  the  order  is  one  higher  than  the  power  of  the  initial  con- 
centration to  which  this  time  is  inversely  proportional. 

249.  (C)  Suppose  the  change  in  concentration  x  is  plotted 
against  the  time  t.  The  velocity, 

dx 
U=~di 

at  any  point,  is  the  tangent  of  0,  the  slope  of  the  curve. 
(Draw  a  short  tangent  to  the  curve  at  this  point.  Make  this 
line  the  hypothenuse  of  a  right  triangle  whose  sides  are  parallel 
to  the  axes.  Tan  (f>  is  the  ratio  of  the  vertical  side  to  the  hori- 
zontal, but  the  vertical  side  is  proportional  to  the  small 
change  dx  and  the  horizontal  side  is  proportional  to  the  cor- 
responding small  change  dt.) 

Let  UT  be  the  velocity  where  the  change  is  xt  and  u2  the 
velocity  where  the  change  is  x2. 

u^  =  k(c—x^)n     U2  =  k(c—x2)n 
Take  logarithms  of  both  sides  and  subtract 

log  Ut-log  u2  =  n[log  (c-xj-log  (c-x,)] 


-    -  (165) 

!  C-X* 

log  - 
& 


REACTION    BETWEEN    HYDRIODIC   AND    BROMIC   ACIDS.       243 

EXPERIMENT  XXXV. 

Determination  of  the  Order  of  Reaction  Between  Hydriodic  Acid  and 

Bromic  Acid.* 

An  aqueous  solution  of  these  two  acids  will  be  obtained  by  dis- 
solving their  potassium  salts  in  acidulated  water  and  the  progress 
of  the  reaction  will  be  followed  by  measuring  the  iodine  set  free. 
The  order  of  the  reaction  will  be  determined  by  method  (B). 

Prepare  the  following  apparatus:  A  thermostat  at  about  25°. 
Four  100  c.c.,  one  300  c.c.,  and  one  500  c.c.  Erlenmeyer  flasks. 
A  burette  with  n/ioo  sodium  thiosulphate  solution.  Fresh  starch 
solution. 2  One  25  c.c.  pipette.  Prepare  decinormal  solutions  of 
potassium  iodide,  potassium  bromate,  and  hydrochloric  acid. 

Put  25  c.c.  of  the  potassium  iodine  solution  in  the  300  c.c.  flask 
and  25  c.c.  of  the  potassium  bromate  solution  in  one  of  the  100  c.c. 
flasks  and  set  them  both  in  the  thermostat.  Pour  100  c.c.  of  the 
hydrochloric  acid  solution  and  100  c.c.  of  water  in  the  300  c.c.  flask. 
Into  each  of  the  other  small  flasks  put  50  c.c.  of  cold  water. 

When  the  reagents  have  attained  the  temperature  of  the  ther- 
mostat, note  the  time,  pour  the  bromate  solution  into  the  300  c.c. 
flask,  mix  thoroughly  and  immediately  pipette  25  c.c.  of  the  mixture 
into  one  of  the  small  flasks  of  cold  water,  noting  the  time  when  the 
reaction  is  stopped  by  the  cold  water.  Add  starch  solution  and  titrate 
with  the  .01  n  thiosulphite.  After  about  three  minutes,  withdraw 
2  5  c.c.  more,  note  the  time  of  mixing  with  the  cold  water,  and  titrate. 
Make  the  next  interval  about  ten  minutes  and  gradually  increase 
the  interval.  The  last  interval  should  be  about  an  hour.  Allow 
the  solution  to  stand  a  day  before  the  final  analysis. 

When  the  time  intervals  are  sufficiently  long,  commence  a  second 
similar  experiment  with  half  the  concentrations.  The  500  c.c. 
flask  (in  the  thermostat)  contains  initially  25  c.c.  potassium  iodine 
solution,  225  c.c.  hydrochloric  acid  solution,  and  225  c.c.  water. 
25  c.c.  of  the  bromate  solution  are  added  from  a  flask  which  has  also 
been  standing  in  the  thermostat  and  all  the  observations  are  repeated. 

Plot  the  observations  in  two  curves,  with  times  as  abscissae  and 
titrations  as  ordinates.  From  the  two  curves,  find  the  ratios  of  the 
times  required  for  one-half,  one-third  and  two-thirds  the  total  change, 
as  represented  by  the  titrations,  and  find  the  means.  If  the  ratio 
is  unity,  that  is,  if  the  time  for  a  definite  fraction,  is  independent  of 
the  concentration,  we  know  the  reaction  is  of  the  first  order.  If  the 
times  are  inversely  proportioned  to  the  concentrations,  we  know  it  is 
of  the  second  order,  etc. 

Write  the  equation  for  the  reaction  in  accordance  with  your 
conclusion  respecting  its  order. 3 

QUESTIONS. 

1.  Choose   three  points  on  one  of  the  curves  and   determine   the 
order  by  method  (A).  N»ij 

2.  Choose  two  points  on  one  of  the   curves  and  determine  the 
order  by  method  (C). 

1  Ostwald,  Zeit.  phys.  Chem.,  1888,  ii,  p.  127. 

2  Kahlbaum's  soluble  starch,  in  paste  form,  is  extremely  convenient  since  it 
dissolves  in  cold  water  and  keeps  well. 

3  See  discussion  by  Ostwald,  Zeit.  phys.  Chemie,  1888,  ii,  p.  127. 


244  CHEMICAL    KINETICS. 

250.  Determination  of  the  Order  of  One  Component  of  a 
Reaction. — If,  as  is  often  the  case,  the  number  of  reacting 
molecules  of  the  different  substances  is  not  the  same,  the 
order  for  each  kind  of  molecule  may  be  determined  separately 
by  using  such  high  concentrations  of  the  other  substances  that 
they  remain  practically  constant.  If  the  initial  concentra- 
tion of  the  component  whose  order  we  wish  to  determine  is 
Cj,  we  make  the  other  initial  concentrations  c2,  c3,  etc.,  so  great 
that  Equation  150  becomes 

u^kA(cl-x)n  (166) 

where  A  is  the  product  of  the  other  concentrations  raised  to 
their  respective  powers,  and  is  practically  a  constant,  because 
clf  and  hence  x,  is  relatively  very  small.  Therefore,  methods 
(A),  (J5),  and  (C)  are  available  because  it  is  immaterial  if  k 
is  multiplied  by  a  constant  factor. 

PROBLEMS  XVI. 

1.  Jiiptner,  ii,  i,  p.  133,  gives  the  following  figures  for  the  reaction 
between     tenth-normal     solutions     of    ferrous    chloride,    potassium 
chlorate,  and  hydrochloric  acid. 

t  x 

o  o 

5  .0048  Determine    the    order  of 

35  .0238  the  reaction  by  methods 

no  -0452  (A),  and  (C). 

170  .0525 

2.  Prove  that  the  reaction  described  in  Problem  XIV,  3,  is  of  the 
first  and  not  of  the  second  or  third  orders. 

3.  Prove  that  the  reaction  described  in  Problem  XV,  i,  is  of  the 
second  and  not  of  the  first  or  third  orders. 

4.  Determine  the  order  of  the  following  reaction: 

2  Fed,  +  SnCl2  =  2  FeCl2  +  SnCl4 

The  initial  concentration  of  each  of  the  chlorides  on  the  left  was 

.0625. 

Time  Change  in  Concentration. 

i.  -01434 

1.75  .01998 

3-  .02586 

4-5  -°3°76 

7.  .03612 

n.  .04102 

25.  .04792 

40.  .05058 

(Meyer,  p.  32.) 


ORDER    OF    ONE    COMPONENT.  245 

5.   Determine  the  order  of  the  following  reaction: 

6  FeCl2  +  KC103  +  6  HC1  =  6  FeCl3  +  KC1  +  3  H2O 

When  the  initial  concentration  of  all  three  substances  on  the  left 
was  o.i  the  changes  in  concentration  were  as  follows: 

Time  Change  in  Concentration. 

5  .0048 

15  .0122 

35  -0238 

60  -0329 

no  -0452 

170  .0525 

(Meyer,  p.  33.) 

EXPERIMENT  XXXVI. 

Determination  of  the  Order  of  the  Reaction  of  Potassium  Ferri- 
cyanide  when  Reacting  with  Potassium  Iodide.1 

We  shall  make  the  concentration  of  the  potassium  iodide  so  great 
compared  with  that  of  the  ferricyanide,  that  the  concentration 
of  the  former  may  be  regarded  as  constant  and  therefore  we  can  use 
Equation  166  and  methods  (A),  (B),  and  (C). 

Prepare  the  following.  Burette  supplied  with  n/ioo  sodium  thio- 
sulphate  solution.  Fresh  starch  solution,  n/^o  potassium  ferricy- 
anide solution,  n/2  potassium  iodide  solution,  25  c.c.  pipette,  100  c.c. 
Erlenmeyer  flask.  If  the  temperature  of  the  room  is  likely  to  vary 
much,  a  thermostat  at  about  2  5°  should  also  be  provided. 

Noting  the  time  of  mixture,  pipette  2  5  c.c.  of  each  solution  into 
the  Erlenmeyer  flask,  mix  thoroughly  and  add  a  little  starch  solution. 
The  solution  will  soon  show  color  due  to  the  liberated  iodine.  After 
about  five  minutes  add  thiosulphite  solution  from  the  burette  until 
the  blue  color  disappears  and  record  the  amount  and  the  mean  time 
of  the  titration.  About  ten  minutes  later  add  more  thisulphite 
until  the  color  disappears  again,  note  the  amount  and  the  time,  and 
so  continue,  increasing  the  intervals  of  time.  The  final  titration 
should  be  two  days  after  the  initial  mixing. 

Plot  thiosulphite  titrations  against  the  time  and  draw  a  smooth 
curve.  First  apply  method  (A),  using  Equations  154,  159,  and  163' 


c-x  '    c-x  '      (c-x)* 

are  mere  ratios,  and  therefore  c  and  x  in  these  expressions  can  be 
expressed  in  any  units.  We  will  therefore  substitute  the  final 
titration  for  c,  and  for  x  we  will  substitute  the  titration  at  the 
corresponding  time  t.  Where  c  occurs  outside  of  these  ratios,  the 
actual  value  (1/80)  must  be  substituted.  In  this  manner  calculate 
the  velocity  constant  for  three  points  on  the  curve  by  each  formula. 
If  one  of  the  formulae  gives  values  which  agree  closely,  we  can  assume 
that  the  reaction  is  of  the  corresponding  order  as  far  as  the  ferri- 
cyanide is  concerned.  State  your  conclusions. 

1  Donnan  and  Le  Rossignol,  Journ.  Chem.  Soc.,  1903,  Ixxxiii,  p.  703. 


246  CHEMICAL    KINETICS. 

If  time  permit,  repeat  the  experiment  with  .in  potassium  ferri- 
cyanide  solution  and  n/4o  potassium  iodide  solution,  and  thus  deter- 
mine the  number  of  molecules  of  potassium  iodide  taking  part  in 
the  reaction. 

QUESTIONS. 

1.  Choose  two  points  on  the  curve  and  calculate  the   order    by 
method  (C). 

2.  The  thiosulphite  is  added  to  the  entire  solution  rather  than  a 
small  sample  because  an  excess  of  free  iodine  disturbs  the  reaction. 
What  other  error  is  thereby  introduced?     Estimate  its  magnitude. 

251.  Incomplete  Reactions.     Equilibrium  Constant. — In  the 

case  of  many  reactions,  possibly  in  all  reactions,  the  chemical 
action  ceases  before  the  change  is  complete.  The  equilibrium 
finally  obtained  is  undoubtedly  kinetic,  that  is,  the  original 
action  does  not  cease,  but  is  counteracted-  by  the  reverse 
reaction  of  the  products  formed.  Therefore,  at  equilibrium, 
the  velocity  of  the  reverse  reaction  must  be  the  same  as  that 
of  the  original  reaction. 

If  we  represent  the  reaction  as  (Equation  149) 

n,A  +  n2B  +  nf—  -  =  nJA'  +  n2'B'  +  n3'C' 

c,        c2        c3  cj          ca'          cs' 

with  the  respective  initial  concentration  represented  by  the 
letters  beneath,  and,  if  equilibrium  is  established  after  the 
change  of  concentration  on  each  side  is  x,  then,  the  velocity 
of  the  original  reaction  (from  left  to  right)  is 


and  the  velocity  of  the  counter  reaction  (from  right  to  left)  is 

u'  =  k'  (c  '  —  x}H1'(c  '  —  x}m' - 

Since  at  equilibrium  the  two  velocities  must  be  equal 

u  =  u' 


-K-      !-.--  (I67) 

k  -      -(c>-xys(c'-xy*'- 

where  K  is  a  constant  called  the  equilibrium  constant.  Ex- 
pressed in  words,  at  equilibrium,  the  ratio  of  the  products  of 
the  concentrations  of  the  reacting  molecules -on  the  two  sides 
of  the  chemical  equation  is  a  constant.  This  theorem  is 


INCOMPLETE    REACTIONS.  247 

often  called  the  Mass  Law.  A  more  formal,  thermodynamic, 
derivation  of  this  equation  will  be  given  later,  together 
with  many  illustrations. 

As  an  illustration  of  an  incomplete  reaction,  we  shall  con- 
sider the  catalysis  of  ethyl  acetate  when  the  water  is  not  in 
great  excess.1 

C2H5OOC2H3  +  H2O  +±  C2HSOH  +  HOOC2H3 

The  sign  ^±  signifies  the  reversibility  of  the  reaction. 
Let  c^  =  concentration  of  water,  c2=that  of  the  ester,  cx'  = 
that  of  the  alcohol,  and  c2f  =that  of  the  acetic  acid.  If  at  any 
time  t,  the  concentration  of  each  substance  has  changed  x, 
the  velocity  of  catalysis  is 

u  =  k  (cl  —  x)  (c2  —  x) 
and  that  of  esterification  is, 

u'  =  kf  (c,f  +  x)  (c2f  +  x} 
The  actual  net  velocity  of  catalysis  will  be 

u  -  u'  =~  =  k  (c,  -  x)  (c3  -x)-k'  (Clf  +  x)  (c2r  +  x)    (i 68) 

At  equilibrium,  let  x=~x, 

k'  _          (Cl-x)(c2-x)  ,} 

T  •>,'+*)(*,'+*) 

25 ia.  We  wish  to  determine  k  and  k' '.  An  obvious  method 
is  to  plot  the  change  in  concentration  x  against  the  time  /,  and 
determine  the  net  velocity,  for  a  particular  value  of  x,  from 
the  slope  of  the  curve  (§29).  Their  ratio  can  be  determined 
from  the  final  concentrations  and  thus  both  can  be  found. 

It  is  also  not  difficult  to  integrate  Equation  168,  but 
this  general  result  is  not  of  particular  interest.  If  all  four 
substances  are  initially  present  in  appreciable  amounts,  the 
reaction  at  ordinary  temperatures  is  far  too  slow  for  a  labo- 
ratory exercise. 

One  case,  however,  offers  no  experimental  difficulties  and 

1  Knoblauch,  Zeit.  phys.  Chem.,  xxii,  p.  268;  Van't  Hoff,  i,  p.  200. 


248  CHEMICAL    KINETICS. 

we    shall    integrate    the    above   equation   for   the  particular 
conditions  of  this  case. 

Let  one  of  the  substances  on  each  side  be  originally  present  in 
large  excess,  let  one  of  the  other  substances  be  present  in  com- 
paratively small  excess,  and  let  the  fourth  substance  be  absent. 
Let  d  and  d'  be  so  large  that  x,  in  comparison,  is  negligible; 
let  c2  be  small  and  c2'  =zero.  The  velocity  at  any  time  t  is 

_  dx  _  .  .          w          ,      ,,  ,    ,       ,  (       , 

lir   —   T"""    *&    \&  I    ~~  %)      \G  2  %)  "^        \^I       "T"   "^)  ^  \  ^   ^9  / 

=  k[d(c2-x)-Kd'x] 


dx 


ctc2-(d  +  Kd')x 

o 


k'  =  Kk 

The  corrected,  equilibrium,  values  of  cl  and  c/  should  be  used 
in  calculating  K  by  equation  167'. 

PROBLEMS  XVI. 

1.  A  bottle  contains  500  c.c.  of  50%  alcohol.      If  5  grams  of  ethyl 
acetate  are  added,  how  many  grams  of  acetic  acid  must  also  be  added 
to  preserve  the  equilibrium?     (The   equilibrium   constant   of  ethyl 
acetate  is  4.) 

2.  What  should  be  added  to  the  former  solution  to  (a)  increase, 
(b)  decrease  the  amount  of  ester? 

3.  When  equivalent  amounts  of  methyl  acetate  and  acetic  acid 
are  mixed,  67.5%  of  each  disappear.     Calculate  K. 

4.  Hydriodic  acid  partially  dissociates  at  moderately  high  tempera- 
tures,   (2HI=H2+I2).      Meyer  and   Bodensteini  found  that  at  440° 
K  =  .02  and,  also,  the  following  figures  are  taken  from  this  work. 

t   (hours)  x 

5  .023  5       Plot,  and  determine  k  and  k' 

15  -°755       by  the  first  method  of  §2 5ia. 

60  .19 

EXPERIMENT  XXXVII. 
Equilibrium  of  Ethyl  Acetate,  Water,  Alcohol  and  Acetic  Acid. 

We  shall  mix  a  large  amount  of  water  clt  a  large  amount  of  alcohol 
Ci',  and  a  small  amount  of  ethyl  acetate,  c2.  We  shall  determine 
the  velocity  of  the  reaction  and  the  final  equilibrium  concentra- 

1  Berichte,  1893,  xxvi,  p.  1146;  Zeit.  phys.  Chem.,  1894,  xiii,  p.  56. 


ETHYL  ACETATE,  WATER,  ALCOHOL  AND  ACETIC  ACID.       249 

tion,  and,  substituting  in  Equations  167' and  170,  we  shall  determine 
k,  the  velocity  constant  of  catalysis,  and  k',  the  velocity  constant 
of  esterification. 

The  apparatus  required  and  the  proceedure,  are  the  same  as  in 
Experiment  XXIX,  but  the  initial  mixture  consists  of  25  c.c.  n/io 
hydrochloric  acid,  25  c.c.  ethyl  alcohol,  and  3  c.c.  of  ethyl  acetate. 
The  titrations  should  be  expressed  in  concentration  of  acid  by  multi- 
plying the  titre,  per  c.c.  of  solution,  by  the  standardized  concentra- 
tion of  the  baryta  solution.  These  concentrations  should  be  plotted 
against  the  time  and  a  smooth  curve  drawn.  A  second  curve 
should  be  drawn  to  give  the  concentration  of  acetic  acid.  This 
curve  will  be  below  the  other  curve,  an  amount  equal  to  the  con- 
centration of  hydrochloric  acid.  The  latter  may  be  taken  as  the 
initial  ordinate  of  the  first  curve,  or  calculated. 

The  first  calculation  is  that  of  K  by  Equation  167'.  The  initial 
concentration  of  water,  alcohol,  and  ethyl  acetate  should  be  deter- 
mined from  the  amount,  total  volume,  density,  and  molecular  weight. 
For  example,  the  initial  concentration  of  ethyl  acetate  is 

3X. 905X1000^ 
88X53 

The  final  concentration  of  acetic  acid,  and  the  change  in  con- 
centration of  the  other  three  components,  is  the  final  ordinate  of 
the  second  curve. 

Having  calculated  K,  select  three  typical  points  on  the  curve 
and  for  each  determine  k  by  Equation  170.  From  the  mean  value, 
and  K,  calculate  k'  by  Equation  167  (first  part). 


QUESTIONS. 

1.  If    we    substitute    total    change    in    concentration    for    initial 
concentration  and  consider  the  actual  velocity  constant   as  k — k', 
the  reaction  behaves  as  approximately  monomolecular.      Determine 
in  this  manner  an  approximate  value  for  k — k'. 

2.  One  hour  after  the  beginning  of  the  reaction,  what  was   (a) 
the    concentration   of   each    component    (use    curve)  ?      (6)    velocity 
of  catalysis?      (c)  velocity  of  esterification?      (d)  net  velocity? 


CHAPTER  VII. 
CHEMICAL  STATICS.1 

252.  Equilibrium.  Variation  of  Equilibrium  with  Tempera- 
ture.2— We  have  already  (§251)  derived,  from  considerations 
of  the  kinetic  equilibrium,  the  theorem  that  the  ratio  of  the 
products  of  the  concentrations  on  the  two  sides  of  a  chemical 
equation  is  constant,  at  equilibrium.  We  shall  now  deduce 
this  same  "mass  law"  from  a  consideration  of  the  mechanical 
work  and  heat  involved  in  a  chemical  equation  in  order  that 
we  may  find  how  the  equilibrium  varies  with  the  temperature. 

To  make  our  ideas  definite  we  shall  consider  a  particular 
reaction,  namely, 

2H2  +  O,<=»2H2O 

The  sign  <=^  is  used  because  we  shall  consider  that  the  three 
substances  are  present  in  their  equilibrium  proportions.  At  a 
high  temperature,  the  concentrations  of  all  three  would  be 
of  the  same  order  of  magnitude.  We  shall  devise  a  mechanical 
process  which  will  carry  on  the  above  reaction  from  left  to 
right,  a  device  often  used  in  physical  chemistry. 

Suppose  the  mixture  is  under  considerable  pressure  in  a 
large  gasometer,  which  has  three  openings  closed  by  slides 
(Fig.  84).  Each  opening  is  covered  by  a  diaphragm,  one  dia- 
phragm is  only  permeable  to  hydrogen,  another  is  only  per- 
meable to  oxygen,  and  the  third  allows  only  water  vapor  to 
pass.  (Compare  the  permeability  of  hot  paladium  to  hydrogen , 
of  copper  ferrocyanide  to  water,  etc.).  Let  the  equilibrium 
concentration,  the  partial  pressure,  and  the  volume  occupied 
by  one  gram  molecule  be,  for  hydrogen  clt  plt  vlt  respectively; 
for  oxygen  c2,  p2,  v2,  and  for  water  vapor  c,  p,  v.  Suppose  we 
also  have  three  great  reservoirs  containing  these  three  gases  at 

1  General  references  for  this  chapter  are  given  with  those  for  Chap.  VI. 

2  Van't  Hoff,  i,  pp.  98-103;  Jiiptner,  ii,  Chap.  I;  Nernst,  iv,  Chap.  III. 

250 


EQUILIBRIUM. 


251 


IH,, 


H20j 


io2 


unit  concentration  (one  gram  molecule  per  litre).  By 
Avogadro's  law,  the  pressure  in  all  three  reservoirs  will  have 
the  same  value  P.  Let  V  represent  the  volume  of  one  gram 
molecule,  i.e.,  one  litre.  The  temper- 
ature of  the  entire  system  is  kept 
constant.  Having  placed  a  suitable 
cylinder,  with  piston,  tight  against 
the  diaphragm  permeable  to  water 
vapor,  draw  the  slide,  and  allow  the 
piston  to  move  out  so  slowly  that 
the  pressure  in  the  cylinder  is  always 
practically  p.  Allow  two  gram 
molecules  to  leave.  Then  close  the 
slide  and  the  cylinder,  and  allow  the 
piston  to  move  out  still  more  until 
the  pressure  has  fallen  to  P.  Now 

connect  the  cylinder  with  the  water  vapor  reservoir,  and 
slowly  moving  the  piston,  force  all  the  water  vapor  into  the 
reservoir.  The  gasometer  and  reservoirs  are  supposedly  so 
large  that  the  addition  or  removal  of  two  gram  molecules  does 
not  appreciably  change  the  pressure. 

253.  Let  us  calculate  the  work  done  by  the  gas.     In  the 
firs,t  step  the  work  done  is  pv  (Equation  62) ;  in  the  next  it  is 

2V  2V 


FIG.  84.' 


/**-*»/* 

2V  2V 

>r,  by  the  gas  law  (Eq.  57) 

pv  =  RwO  .'.  p 


2Rd\n 


[n  the  final  compression  into  the  reservoir  the  work  done  by 
te  gas  is  —PV.  (The  negative  sign  signifies  work  done  upon 
ie  gas.)  But  since  the  temperature  and  mass  of  the  gas  are 
mstant  pv=PV,  and  therefore  the  total  work  done  by  the 

>ater  vapor  is 


252  CHEMICAL    STATICS. 

By  means  of  a  similar  cylinder  and  piston,  we  will  take  two 
gram  molecules  of  hydrogen  from  the  hydrogen  reservoir,  at 
constant  pressure  P,  compress  it  to  pressure  plt  and,  removing 
the  slide,  force  it  into  the  gasometer  through  the  diaphragm 
permeable  to  hydrogen.  The  work  done  upon  the  gas  will  be 


We  shall  introduce  similarly  one  gram  molecule  of  oxygen, 
and  the  work  done  upon  the  gas  will  be 


When  the  water  vapor  was  removed,  the  hydrogen  and 
oxygen  formed  more  water  vapor  to  maintain  the  equilibrium 
and  this  hydrogen  and  oxygen  were  replaced  by  that  introduced. 
The  initial  and  final  conditions  of  the  cylinder  are  therefore 
identical.  Two  gram  molecules  of  water  vapor  have  been 
formed  from  hydrogen  and  oxygen.  The  total  amount  of 
work  done  by  the  gas  is 


(Twice  the  logarithm  is  the  logarithm  of  the  square.) 

Since  the  volumes  are  inversely  as  the  concentrations  and  the 
concentration  of  the  reservoirs  is  unity,  we  may  also  write 
this 

(172) 


Where  K  is  our  former  equilibrium  constant;  that  is,  the 
product  of  the  concentrations  of  the  reacting  molecules  on 
one  side  of  the  equation  divided  by  the  product  of  the  con- 
centrations on  the  other  side.  Since  the  water  and  hydrogen 
molecules  occur  twice,  their  concentrations  are  squared. 

254.   By  Equation  101  in  the  chapter  upon  thermodynamics 


APPLICATIONS    OF    THE    MASS    LAW.  253 

/  is  the  decrease  in  internal  energy  produced  by  the  reaction, 
id  is  evidently  the   heat  emitted  by  the  reaction  when  no 

external  work  is  done  and  which  we  have  represented  by  —  Q 

calories  or  —  JQ  ergs  (§§129,  184) 

dW      ™    T>  .  „„  dlnK- 

dlnK 


.'.  RfflnK  +JQ  = 


dO 


dO 

his  is  Van't  Kofi's  "isochore"  equation. 
Notice  that  —  Q  has  been  taken  as  the  heat  emitted  in  the 
ormation  of  the  substance  whose  concentration  appears  in  the 
umerator  of  K  and  the  number  of  gram  molecules  is  the  num- 
er  of  molecules  appearing  in  the  chemical  equation.     For 
example,  —  Q  is  the  heat  of  formation  T  of  two  gram  molecules  of 
water.     Since  the  amount  of  water,  and  therefore,  K,  decreases 
with  rise  of  temperature,  Q  is  negative,  or  heat  is  emitted  when 
water  is  formed  from  hydrogen  and  oxygen. 

Equation  173  tells  us  that  if  the  temperature  is  constant, 
In  K  and  hence  K  is  constant  and  thus  Van't  Hoff  derived 
the  mass  law  which  Guldberg  and  Waage  derived  from 
nsiderations  of  the  kinetic  equilibrium. 

Applications  of  the  Mass  Law. 

255.  For   convenience    of   reference,    we    will    restate    the 
general  expression  for  the  mass  law  (Equation  167) 


cI  c2  ---  are  the  equilibrum  concentrations,  and  n,  n2  — 
are  the  numbers  of  the  different  kinds  of  molecules  on  one  side 
of  the  reaction  and  the  primed  letters  refer  to  the  other  side. 
If  we  increase  a  concentration  in  the  numerator,  some  of  the 
1  Remember  that  positive  heat  of  formation  =  heat  emitted  (Chap.  IV). 


254 


CHEMICAL    STATICS. 


other  concentrations  must  decrease  or  some  of  those  in  the 
denominator  must  increase. 

256.  Equilibrium  of  Electrolytes.1 — Salts,  acids,  and  bases, 
in  aqueous  solution  partially  dissociate  into  particles  called 
ions,  which  possess  electrical  charges.  This  phenomenon, 
which  has  been  mentioned  previously,  will  be  considered  later 
in  detail. 

Suppose  we  dissolve  a  weak  acid,  such  as  acetic  acid,  in 
water.  It  will  dissociate  according  to  the  equation 

CaH3OOH<=»Q(H3OO'  +  H 

C  C,  C2 

(The  dot  signifies  a  positive  charge  and  the  prime  a  negative 
charge.) 


Add  a  little  hydrochloric  acid 


The  increase  in  c2  will  require  a  decrease  in  clt  and  a  cor- 
responding increase  in  c,  or,  the  addition  of  hydrochlori< 
acid,  because  it  has  a  common  ion,  decreases  the  dissociatic 
and  hence  the  strength  of  the  acetic  acid  (§240).  The  acetic 
acid  will  reciprocally  decrease  the  dissociation  of  the  hydro- 
chloric acid,  but  this  strong,  highly  dissociated  acid  will 
be  affected  to  a  far  smaller  degree.  The  addition  of  sodium 
acetate  would  increase  GI  and  similarly  reduce  the  strength 
of  the  acetic  acid. 

257.  Consider  a  solution  of  a  weak  base 


Walker,  Chap.  XXVI;  Mellor,  Chap.  IX;  Nernst,  iii,  Chap.  IV. 


DISSOCIATION    OF    WATER.  255 

The  addition  of  a  strong  base,  such  as 
NaOH^OH'  +  Na- 
c'  c,          c2f 

increases  ct  and  therefore  decreases  c2  and  increases  c,  thus 
decreasing  the  dissociation  and  strength  of  the  weak  base. 
The  strong  base  will  be  affected  to  a  far  less  extent.  The 
addition  of  a  salt,  such  as  ammonium  chloride,  increases  c2, 
and  therefore  has  a  similar  effect. 

If  the  dissolved  substance  is  only  slightly  dissociated  and 
the  concentration  is  near  saturation,  the  addition  of  one 
of  the  ions  may  necessitate  not  only  the  reduction  in  dis- 
sociation, but  the  increase  in  undissociated  salt  may  be  so 
great  that  the  solution  may  become  supersaturated  and  the 
substance  may  be  thrown  out  of  the  solution,  or  the  solubility 
decreased.  The  best  example  of  this  is  the  so-called  "salting 
out  of  soap."  When  sodium  chloride  is  added  to  a  solution  of 
the  slightly  dissociated  sodium  salts  of  the  higher  fatty  acids, 
the  increase  in  the  concentration  of  the  sodium  ions  so  greatly 
increases  the  concentration  of  the  undissociated  fatty  acid 
salts,  that  the  saturation  concentration  is  exceeded  and  they 
are  precipitated. 

258.  Dissociation  of  Water.  lonization  Constant.1  —  Pure 
water  suffers  electrolytic  dissociation  to  a  very  slight  extent. 


C  Ct  C2 

r,  the  concentration  of  the  water,  is  practically  constant. 


(175) 


K'  is  sometimes  called  the  ionization  constant  (avoid  confusion 
with  the  more  common  definition  in  §301).  Kf  will  be  de- 
termined in  Experiment  LVI.  For  pure  water  its  value  is 
about  io-J4  (§328)  and 


1  Jtiptner,  ii,  pp.  76-79;  Walker,  pp.  314-316. 


256  CHEMICAL    STATICS. 

Since  Kf  is  very  small,  the  concentration  of  OH'  ions  in  an 
acid  solution  is  exceedingly  low  and  the  same  is  true  of  H* 
ions  in  an  alkaline  solution. 

259.  Indicators.1 — The    mass   law   explains   the   action   of 
indicators.     For  example,  phenolphthalein  is  a  weak  acid  with 
a  pink  negative  ion.     The  pink  color  therefore  appears  when 
the  concentration  of  the  negative  ions  is  appreciable,  and  the 
mass  law  tells  us  that  this  requires  that  the  concentration  of 
the  hydrogen  ion  shall  be  very  small,  or  the  solution  must  be 
alkaline.     If  the  concentration  of  the  hydrogen  ions  is  less 
than  io-9  normal,  the  pink  color  appears;  if  it  is  greater  than 
io-8  normal,  the  color  is  absent. 

260.  Dilution    Law.     Dissociation    or    Affinity    Constant.2 
Degree   of   Dissociation. — Suppose  we   have  a  solution  of  a 
weak  acid,  base,  or  salt,   for  example,  acetic  acid,  and  the 
concentration  is  such  that  one  gram  equivalent  is  dissolved  in 
V  litres. 

C2H3OOH<=»C2H3OO'  +  H- 
c.  Ct  c2 

The  fraction  of  the  total  acid  dissociated,  or  the  degree  of 
dissociation,  we  shall  represent  by  a. 

a 


" 


V  V 

cx^  a2 


The  law  expressed  in  this  equation  was  discovered  by 
Ostwald.3  It  is  often  known  as  his  dilution  law.  In  Experi- 
ment L  we  shall  prove  that  K  is  very  approximately  constant, 
however  we  may  vary  V.  K  is  called  the  dissociation  constant. 
It  is  also  often  called  the  affinity  constant,  because  all  th< 
properties  which  measure  the  strength  of  an  acid  or  base  are 
proportional  to  the  dissociation  (concentration  of  H',  or  OH' 
ions,  §240).  This  subject  is  discussed  further  in  §§299,  300. 

1  Walker,  pp.  317,  323,  338;  Mellor,  pp.  215-216;  Nernst,  pp.  489-491. 

2  Whetham,  pp.  341-346;  Walker,  pp.  244-248;  Mellor,  pp.  189-193. 

3  Ostwald,  Zeit.  phys.  Chem.,  1887,  ii,  p.  36. 


HYDROLYSIS. 


257 


261.  Hydrolysis.1 — We    can    now    explain    the    acidity   or 
alkalinity   in    the    hydrolysis   of   aqueous   solutions   of   salts 
composed  of  a  strong  acid  and  a  weak  base,  or  vice  versa. 
Take  ferric  chloride,  for  example.     The  equation  for  its  disso- 
ciation is 

FeCl3  +  3  HaO<=±Fe  +  3  Cl'  +  3  IT  +  3  OH" 

The  dissociation  constant  of  HC1  is  so  much  greater  than  that 
of  Fe(OH)3  that  the  number  of  H'  ions  is  vastly  greater  than 
the  number  of  OH'  ions,  or  the  solution  behaves  as  an  acid.  If 
the  base  is  the  stronger,  the  hydroxyl  ions  are  in  great  excess. 

262.  Gaseous  Dissociation.     Calcium  Carbonate.2 — At  mod- 
erately high  temperatures,  calcium  carbonate  partially  dis- 
sociates into  calcium  oxide  and  carbon  dioxide  gas,  according 
to  the  equation 

CaCO3<=»CaO  +  CO2 


Since  both  the  carbonate  and  oxide  are  solids,  cz  and  c2  are 
practically  constant,  and  therefore  the  concentration  of  the 
carbon  dioxide  is  constant.  In  other  words,  if  the  temperature 
remains  constant,  the  pressure  of  the  carbon  dioxide  gas  is 
independent  of  the  amount  of  calcium  oxide  and  carbonate 
>resent. 

263.  Nitrogen  Peroxide.3 — Nitrogen  peroxide  exists  in  two 
molecular  aggregations,  the  equilibrium  between  which  may 
be  determined  by  the  mass  law. 


N204: 

C 


2N02 

C-, 


1  See  reference  for  §242. 

2  Le  Chatelier,  C.  R.,  1883,  102,  p.  1243. 

3  Natanson,  Wied.  Ann.,  1885,  xxiv,  p.  454;  1886,  xxvii,  p.  606. 


CHEMICAL    STATICS. 


2S8 

Let  a  =  degree  of  dissociation   (§260)   and  let   V=  volume 
occupied  by  one  gram  molecule  (92  grams) 


of 


V  (i  -a) 
or  Equation  176. 

PROBLEMS  XVIII. 

1.  Derive  Equation  (173)  from  (172)  by  consideration  of  a  thermo- 
dynamic  cycle  between  temperature  6,  where  the  reaction  and  work 
take  place   according  to   Equation  (172),  and  6-dO,  where  the  same 
reaction  occurs  in  the  opposite  direction. 

2.  (a)  Above  what  concentration  of  OH'  ions  is  the  pink  ion  of 
phenolphthalein  visible?     Below  what  concentration  is  it  invisible? 
(§§258,  259.) 

3.  The    degree   of   dissociation   of   a   normal   solution   of   sodium 
chloride  at  18°  is  .678.      What  is  the  dissociation  constant? 

4.  The    dissociation    constant    of    acetic    acid     is     i.8Xio-s.      If 
a  litre  of  water  contains  i  gr.  of  the  acid,   (a)   how  much  is  disso- 
ciated?     (6)    What   would   be   the   approximate   dissociation  if  the 
litre  of  solution  also  contains  100  gr.  of  hydrochloric  acid?  («  is  so 
small  that   it  may  be  neglected  in  comparison  with  unity,  and  Eq. 
194  may  be  used  in  place  of  Eq.  176.) 

5.  The  pressure   of  carbon  dioxide  over  calcium  carbonate  and 
oxide  at   740°  is  25.5  cm.     The  density  of  calcium  oxide  is  3.20, 
that  of  calcium  carbonate  is  2.72.    Calculate  K.      (First  calculate  the 
concentration  of  the 


The  concentration  of  the  CaO  may  be  taken  at     32°C  — ,  etc.) 

56 

EXPERIMENT  XXXVIII. 
Variation  in  the  Dissociation  of  N2O4  with  Change  of  Volume. 

The  gas  tube  of  a  Boyle's  law  apparatus  is  filled  with  dry  nitrogen 
peroxide,  and  the  pressure  is  observed  for  different  volumes,  the 
temperature  being  maintained  constant  by  a  water-bath.  The 
apparatus  is  illustrated  in  Figs.  42  and  420.  The  graduated  tube, 
A,  is  removed  from  the  slide  and  supported  in  front  of  the  mirror 
glass  of  the  slide.  It  is  surrounded  by  a  water  jacket  in  which  there 
is  a  stirrer  F,  and  a  thermometer  T.  The  water  jacket  is  covered 
by  an  asbestos  jacket  (not  shown)  with  narrow,  vertical  windows. 
A  and  C  are  mounted  on  vertical  slides  separated  by  a  vertical 
scale,  upon  which  the  mercury  levels  and  the  bottom  of  the  stopper 
D  are  read  by  means  of  the  sliding  index  /.  A  strip  of  mirror  glass 
behind  A  and  C  will  reduce  error  from  parallax  in  reading. 


THERMAL    DISSOCIATION. 


259 


A  is  filled  with  dry  nitrogen  peroxide  in  the  following  manner: 
About  20  gr.  of  lead  nitrate  are  placed  in  a  15  cm.  side-neck  test- 
tube,  which  is  closed  by  a  paraffined  stopper  and  connected  through 
a  drying  tube,  with  a  gas  condensation  vessel  immersed  in  a  freezing 
mixture  of  ice  and  salt.  (The  gas  condenses  at  —  n°.)  The 
delivery  tube  of  the  condensation  vessel  is  also  supplied  with  a 
drying  tube  to  which  is  connected  a  rubber  tube  leading  to  a  hood 
or  a  window.  A  convenient  form  of  vessel  is  described  in  §84,  but 
a  side-neck  test-tube,  with  a  tube  passing  through  a  paraffined 
stopper  and  reaching  almost  to  the  bottom,  will  answer. 

The  lead  nitrate  is  heated  until  a  considerable  volume  of  the  dark 
brown  gas  has  been  produced  and  a  considerable  excess  has  escaped 
over  what  is  condensed.  The  condensation  vessel  is  disconnected 


80 


C  50 

o 
.240 

0 
i-O 


20 


10°        20°       30°       40°       50°       60°       70°       80° 
Temperature 

FIG.  85. 


from  the  retort,  one  end  is  closed,  and  the  end  of  the  drying  tube 
connected  to  the  other  end,  is  placed  almost  against  the  mercury  in 
A,  which  is  in  the  mean  position.  The  condensation  vessel  is  now 
removed  from  the  freezing  mixture  and  warmed  by  the  hand,  until 
the  tube  A  appears  full  of  the  gas  and  a  considerable  excess  has 
escaped  at  the  top.  The  end  of  the  delivery  tube  is  slowly  removed, 
the  mercury  level  in  A  is  raised  to  within  about  5  cm.  of  the  top  of 
the  tube,  and  the  paraffined  stopper  D  is  forced  in  as  far  as  it  .will 
go.  A  is  then  lowered  about  10  cm.,  and  E  is  filled  with  water  at 
about  60°.  When  the  temperature  has  become  fairly  constant,  the 
temperature  and  the  positions  of  the  mercury  levels  and  the  bottom 
of  the  stopper  are  read  and  recorded.  C  is  now  lowered  about  10  cm., 
the  water  jacket  is  well  stirred,  and  sufficient  hot  water  is  added  to 
raise  the  temperature  to  the  former  value,  and  the  readings  are 


260  CHEMICAL    STATICS. 

repeated.1  C  is  then  lowered  10  cm.  more,  and  the  readings  are 
repeated  and  so  continued,  until  C  is  at  its  lowest  level  and  A — E 
has  been  raised  to  the  highest  possible  position. 

"The  readings  shoulo  be  repeated  at  the  same  temperature.  Read 
the  barometer,  and  calculate  and  tabulate  the  pressures,  and  also  the 
corresponding  volumes  of  the  gas.  Since  the  cross  section  of  A 
is  constant,  length  of  gas  column  may  be  substituted  for  the  actual 
volume. 

Plot  pressures  against  volumes  (or  lengths  proportional  to  vol- 
umes). Find  from  the  curve,  the  volume  VT,  corresponding  to  a 
pressure  of  76  cm.  The  accompanying  diagram 2  (Fig.  85)  gives  the 
degree  of  dissociation  at  different  temperatures  when  the  pressure 
is  76  cm.  If  a  is  tho  degree  ot  dissociation,  and  N  is  the  number  of 
molecules  if  there  were  no  dissociation,  the  actual  number  of  undis- 
sociated  molecules  is  (i  —  «)  N  and  the  number  of  dissociated  mole- 
cules is  2Na.  Therefore,  the  total  number  of  molecules  is  (j  +  a)  N, 
or  the  number,  and  therefore  the  volume,  if  the  pressure  is  constant, 
is  increased  in  the  proportion  i  +a 

Divide  vl  by  (i  +  a)  and  thus  find  what  would  be  the  volume  Vi 
jf  there  were  no  dissociation,  and  locate  this  point  on  the  line  of 
76  cm.  pressure.      By   assuming  Boyle's   law,  find   the    volume   v,' 
if  there  were  no  dissociation  and  the  pressure  was  66  cm 

76  vl'  =  66v2' 

Thus  locate  about  six  points  between  the  highest  and  lowest  pres- 
sures and  draw  a  smooth  dotted  curve  between  them.  For  each 
of  these  points  determine  a,  from  the  ratio  (i  +a)  of  the  actual 
volume  to  the  volume  without  dissociation,  and  calculate 

K=       a2 
V(i-a) 

V  is  the  volume  occupied  by  one  gram  molecule  (92  grams).  As 
the  mass  of  gas,  and  the  cross  section  of  the  gas  tube  are  constant, 
and  as  we  are  principally  interested  in  the  constancy  of  K,  we  may 
substitute  for  V  the  actual  volume  or  length  of  the  gas  column. 

QUESTIONS. 

1.  How  could  you  determine  the  degree  of  dissociation  at   dif- 
ferent   temperatures   when    (a)    the    volume    is    constant?      (b)    the 
pressure  is  constant  ? 

2.  What  else  would  you  require  to  determine  the  true  equilibrium 
constant  for  this  temperature? 

264.  The   Distribution   Law.     Partition   Coefficient.— Thus 

far  we  have  only  considered  the  equilibrium  between  molecules 
or  ions  in  one  medium.  If  two  media  are  in  contact,  for  ex- 
ample, liquid  and  vapor  or  two  solvents,  there  will  ultimately 

1  If  the  required  apparatus  is  available,  a  steady  flow  of  water  from  a  ther- 
mostat (§81),  through  the  water  jacket,  is  much  to  be  preferred. 

2  From  table  in  1909  Chem.  Kal.,  ii,  p.  257. 


THE    DISTRIBUTION    LAW. 


26l 


be  kinetic  equilibrium  at  the  separating  surface  for  each 
different  kind  of  molecule.  In  any  interval  of  time,  the 
same  number  of  any  particular  species  of  molecule  will  cross 
from  medium  one  to  medium  two  as  travel  in  the  opposite 
direction.  Therefore,  at  equilibrium  there  must  be  a  fixed 
ratio  between  the  concentrations  in  the  two  media  of  any 
common  molecule.  This  is  called  the  distribution  law  and  was 
discovered  independently  by  Nernst  and  Aulich.1  The 
constant  ratio  is  called  the  partition  or  distribution  coefficient. 

265.  As  a  general  example,  we  shall  consider  the  distribu- 
tion of  benzoic  acid  between  benzol  and  water.  If  it  existed 
in  similar  molecular  form  in  the  two  solvents,  we  should  find 
a  constant  ratio  between  the  concentrations,  as  we  varied  the 
total  amount  of  benzoic  acid  which  was  distributed  between 
the  two  solvents.  We  have,  however,  seen  (§§168,  181)  that 
such  liquids  as  benzol  tend  to  produce  association  of  the 
molecules  dissolved  in  them,  while  water  and  similar  liquids 
produce  dissociation.  If  therefore  a  fraction  /?  of  the  benzoic 
acid  in  the  benzol  is  in  the  form  of  double  molecules,  the 
mass  law  requires  that 


KB 


(177) 


should  be  constant.  For  two  of  the  single  molecules  of  con- 
centration CB(i—jl)  form  one  of  the  double  molecules  of  which 
the  concentration  is  CB{$.  CB  is  the  total  concentration  of 
benzoic  acid  in  the  benzol. 

In  the  water,  where  a  fraction  a  is  dissociated,  we  shall 
have  (Eq.  176) 


__ 

-w  —  /  \ 

(i-a) 
^urther,  if  K  is  the  partition  coefficient, 

—  a) 


(177') 


(177") 


1  Zeit.  phys.  Chem.,  1891,  viii,  p.  105;  p.  no. 


262  CHEMICAL    STATICS. 

for  the  only  common  molecules  in  the  two  solvents  are  those 
which  are  neither  dissociated  nor  associated. 
Combining  Equations  177  and  1  7  7  " 


For  high  concentrations,  ft  is  practically  constant,  a  is 
inappreciable,  and  hence  CW2ICB  is  constant.  If  the  concen- 
trations are  reduced,  ft  decreases  and  a  becomes  appreciable. 
If  a  is  determined  by  one  of  the  methods  described  later,  the 
variation  in  ft  may  be  determined  from  the  variation  in  the  left- 
hand  side  of  the  equation. 

EXPERIMENT  XXXIX. 
Partition  of  Benzoic  Acid  Between  Water  and  Benzol. 

Into  each  of  three  carefully  cleaned  75  c.c.,  labeled,  glass- 
stoppered  bottles  introduce  25  c.c.  of  distilled  water  and  25  c.c. 
of  benzol.  To  the  first  add  2  gr.  of  benzoic  acid,  to  the  second  i  gr. 
and  to  the  third  .5  gr.  Shake  thoroughly  every  minute  for  about 
fifteen  minutes,  and  then  allow  the  solutions  to  rest  undisturbed  for 
five  minutes.  Prepare  a  burette  and  an  approximately  n/2o  baryta 
solution  (§90). 

Pipette  2  c.c.  from  the  benzol  layer  of  the  first  solution,  dis- 
charge into  a  large  excess  of  water  in  a  clean  Erlenmeyer  flask,  add  a 
little  alcohol,  and  titrate  with  the  baryta,  using  phenolphthalein  as 
an  indicator.  The  solution  should  be  vigorously  agitated  throughout  the 
titration.  Find  similarly  the  titrations  for  the  other  two  benzol  layers. 

By  means  of  a  syphon  of  capillary  tubing  carefully  draw  off 
about  10  c.c.  of  the  aqueous  layer  in  the  first  bottle.  Discard  the 
first  cubic  centimeter  or  so  which  may  be  contaminated  by  the 
benzol.  Pipette  5  c.c.  of  the  remainder  into  a  clean  flask  and  titrate. 
Proceed  similarly  with  the  other  water  solutions.  If  carefully 
manipulated,  a  finely  pointed  pipette  may  be  used  in  place  of  the 
syphon.  Reduce  the  titrations  to  a  common  number  of  centi- 
meters, and  tabulate. 

For  each  bottle  calculate  and  tabulate 


CB'    CB  '   CB* 

Since  we  are  only  interested  in  the  constancy  of  these  ratios,  titrations 
may  be  substituted  for  concentrations.  State  which  ratio  is  most 
constant,  and  therefore  what  is  the  probable  molecular  state  in 
each  solvent.  Explain  the  significence  of  any  variation  which  you 
may  find  in  this  ratio. 


VAPORIZATION.  263 

This  experiment  may  be  varied  by  substituting  salicylic  or 
succinic  acid  for  benzoic  acid  and  also  by  substituting  ether  for 
benzol. 

QUESTION. 

1.  Apply  the  distribution  law  to  the  solution  of  a  gas  in  a  liquid. 

2.  Apply    the    distribution    law    to    the    equilibrium    between    a 
saturated  vapor  and  its  liquid. 

Variation  of  Equilibrium  with  Temperature.1 
266.  Van't  Hoff's  equation  (173)  will  be  in  a  more  conven- 
ient  form  if  we  integrate  it   between  the  limits  Kr  and  K2 
and  Ol  and  62. 

K2  02 

JQ    i  dO^_     ^2_JQi_l_      1_ 
0*          K,      R   (0,       02 


/")  /?  /?       /2  \ 

TTTr-  (i79) 


f-«-  ff 

*J  *J 

K,  0, 


We  shall  consider  several  applications  of  this  equation. 

267.  Vaporization. — Let  c=  concentration  of  liquid  (this  is 
practically  constant  for  the  volume  changes  little  for  small 
ranges  of  temperature) ;  let  the  concentration  of  the  saturated 
vapor  =ct  at  absolute  temperature  dlt  and  c2  at  temperature 
62.  If  the  temperature  change  is  small,  we  can  assume  that 
the  vapor  obeys  the  gas  law  (§103). 

pv  =  Rwd 

Since  —  is  the  molecular  concentration, 

v 

C-  =  W'C*  =  W'*'=  ' 

/<(/!  KU2 

etc. 

Q 


.' .   2.302  log  £—Q- 

1  Nernst,  Gott.  Nachricht,  1906,  i,  p.  i;  Van't  Hoff's  Lectures,  i,  p.  136,  149, 
157;  Mellor,  Chap.  XII.      See  also  reference  for  §252. 

2  In  certain  cases  this  equation  gives  results  which  are  in  error  because  Q  is 
not  constant  as  we  have  assumed.     For  a  more  accurate  but  more  complicated 
expression  which  considers  the  variation  in  Q,  see  Nernst,  "Thermodynamics  and 
Chemistry." 


264  CHEMICAL    STATICS. 

Illustration. — Determine  the  mean  latent  heat  of  vaporization  of 
water  between  95°  and  105°. 

From  Regnault's  tables  the  saturated  vapor  pressure  at  95° 
is  633.8  mm.  and  at  105°  it  is  906.4  mm.  Substituting  these  values 
and  the  absolute  temperatures,  Q  comes  out  9120.  Now  Q  was  the 
heat  absorbed  when  no  external  work  was  done,  but  in  the  evaporation 
of  water,  the  work 

pv=RO 

is  done,  where  0  is  the  mean  absolute  temperature  and  v  is  the 
volume  of  one  gram  molecule  of  the  vapor.  (For  the  volume  of  the 
liquid  is  relatively  so  small  that  it  may  be  neglected.)  Adding  Rd  = 
1.985  X  373,  we  have  9860  as  the  mean  latent  heat  for  one  gram 
molecule.  The  ordinary  latent  heat  per  gram  is  this  divided  by 
1 8,  or  548  calories. 


268.  Heat  of  Solution.1 — If  an  excess  of  the  dissolved  sub- 
stance (solute)  is  present  in  the  solution,  there  will  ultimately 
be  kinetic  equilibrium  between  it  and  the  saturated  solution. 
In  any  time  the  same  number  of  molecules  will  go  into  the 
solution  as  return  and  reunite  with  the  undissolved  solute. 
Therefore,  both  the  mass  law  and  Van't  HofFs  equation  are 
applicable.  The  heat  of  solution  per  gram  molecule  replaces 
the  heat  of  formation. 

Let  c  =  concentration  of  undissolved  substance.  This  is 
practically  independent  of  the  temperature,  for  the  coefficient 
of  expansion  is  usually  extremely  small.  Let  CT  =  concentra- 
tion of  the  saturated  solution  at  absolute  temperature  6It  and 
c2  the  concentration  at  02. 

K          Cl     K          °2 
K*=—'K*=  — 

Therefore,  by  Equation  179 

•  \^ \ 

n   s?fi      /)  \ 

(i  80) 


If  the  solubility  increases  with  rise  of  temperature  (the 
ordinary  case)  the  left-hand  side  is  positive  and  Q  is  positive, 
or  heat  is  absorbed.  If,  however,  the  solubility  decreases, 
as,  for  example,  in  the  case  of  calcium  hydrate,  heat  is  emitted. 

1  Jiiptner,  ii,  pp.  219-221;  Nernst,  pp.  552-554. 


HEAT    OF    DISSOCIATION    OF    A    GAS. 


EXPERIMENT  XL. 
Heat  of  Solution  and  Solubility. 


265 


In  a  thermostat  at  2  5°  place  a  glass-stoppered  bottle  containing 
100  c.c.  of  hot  water  and  an  excess  of  benzoic  acid,1  a  5  c.c.  pipette, 
and  an  Erlenmeyer  flask  containing  about  25  c.c.  of  water.  Shake 
the  bottle  vigorously  every  minute  for  fifteen  minutes.  After  allowing 
it  to  rest  undisturbed  for  five  minutes,  pipette  5  c.c.  of  the  clear 
saturated  solution  from  the  bottle  to  the  flask,  and,  removing  the 
flask,  titrate  with  approximately  n/20  baryta  solution  (§90),  using 
phenolphthalein  as  an  indicator.  Repeat  twice  and  take  the  mean. 

Raise  the  thermostat  to  about  65°  and  repeat  the  above  procedure. 
The  pipette  must  be  at  the  temperature  of  the  thermostat  to  avoid 
precipitation. 

The  ratio  of  the  mean  titrations  will  be  the  ratio  of  the  concentra- 
tions. From  this  ratio  and  the  two  absolute  temperatures  calculate 
the  molecular  heat  of  solution. 


QUESTIONS. 

1.  What  is  the  heat  of  solution  per  gram? 

2.  Is   the   solution   of    the   acid    an   endothermic   or   exothermic 
process  ? 

3.  How  would  you  change  the  above  equation   to  allow  for  ion- 
ization? 

269.  Heat  of  Dissociation  of  a  Gas. — If  p  is  the  actual  pres- 
sure of  a  gas  and  p'  is  what  the  pressure  would  be  if  it  occupied 
the  same  volume  but  was  undissociated 

p  =  p'  (z  -f  a]  (181) 

where  a  is  the  degree  of  dissociation.  For  the  number  of 
molecules,  and  therefore  the  pressure,  is  increased  in  the  ratio 
(i+a)  (see  Experiment  XXXVIII).  If  v,  the  volume,  is 
constant  the  total  concentration  is  constant  and  equals  i/v 
By  §263  and  Equation  176 


K  = 


V(i-a) 


.'.  2.30  log 


q2a(i-q)  =    Q    0,-Ot 
ax»(i-aa)      1.98    6,6, 


(182) 


al  is  the  degree  of  dissociation  at  absolute  temperature  dlt 
and  o:2is  that  at  02. 

1  Salicylic,  succinic,  or  boric  acid  may  be  used  in  place  of  benzoic  acid. 


266  CHEMICAL    STATICS. 

PROBLEMS  XIX. 

1.  Prove  that  the  potential  of  any  type  of  molecule  is  the  same 
in  each  of  two  adjoining  media,  when  there  is  equilibrium  (§235). 

2.  Succinic  acid  was  dissolved  in  varying  amounts  in  a  mixture 
of  10  c.c.  of  water  and  10  c.c.  of  ether.     The  following  concentrations 
were  found. 

Water.  Ether. 

.024  .0046 

.070  .013 

.121  .022 

Compare  the  molecular  weights  in  the  two  solvents. 

3.  Benzol  boils  at  80.  i°  at   76  cm.     The  latent  heat  is  93.     Cal- 
culate the  vapor  pressure  at  85°  by  Eq.  179'. 

4.  The  molecular  heat  of  solution  of  succinic  acid  is  6700.      The 
saturation    concentration  at  8.5°  is  4.22%.      Calculate  the  mass  in 
grams  of  a  liter  of  saturated  solution  at  20°. 

5.  The    degree  of  dissociation    of    water  vapor   is    1.89X10-4  at 
1207°  and  18.1X10-4  at  1531°.     Calculate  the  heat  of  dissociation. 

6.  Calculate  the  temperature  centigrade  at  which  10%  of  oxygen 
is    associated    into    ozone,     assuming    the    following    approximate 
figures.     34,100  calories  are  absorbed  when  one  gram  molecule  of 
ozone   is    formed.     Concentration  of   ozone  at   i9io°=.oi%.     (The 
application  of  Eq.  182   is  simpler  if  the  decomposition  of  ozone  is 
considered,  e.g.,  the  degree  of  dissociation  at  1910°  is  .9999;  at  what 
temperature  is  it  .9?) 

7.  The    percentage    dissociation    of    carbon    dioxide    is    .0042    at 
1027°  and  .03  at  1205°.     Calculate  the  heat  of  formation.      (Nernst 
and  Wartenberg.) 

8.  The  equilibrium  constant  of  dibromosuccinic  acid  is  .0000967 
at  15°  and  .0318  at  101°. 

(a)    What  is  the  ratio  of  the  velocities  of  decomposition? 
(6)    What  is  the  heat  of  the  reaction? 

9.  Calculate  the  latent  heat  of  water  at  75°  from  the  vapor  pres- 
sures at   70°  and   80°  (a)    as  given  in  Table  LVII ;  (6)  as  determined 
in  Experiment  IX. 

10.  Calculate  the  heat  of  dissociation  of  N2O4  at  constant  pressure 
(Fig.  85). 

EXPERIMENT  XLI. 

Heat  of  Dissociation  of  Nitrogen  Peroxide,  Variation  of  Dissociation 
with  Temperature.1 

We  shall  use  the  air  thermometer  apparatus  described  under 
Experiment  III.  Fill  the  bulb  with  dry  nitrogen  peroxide  in  the 
following  manner:  Surround  it  by  a  freezing  mixture  and  connect 
it, .through  a  drying  tube,  with  a  condensation  vessel  (§84)  which 
is  also  in  a  freezing  mixture,  and  in  which  nitrogen  peroxide  has  been 
condensed  as  described  in  Experiment  XXXVIII.  Gradually 
remove  the  condensation  vessel  from  the  freezing  mixture,  so  that 
the  nitrogen  peroxide  may  distill  over  into  the  bulb.  Then  remove 
the  bulb  also  from  the  freezing  mixture  and  warm  it  to  about  20°, 
so  that  most  of  the  air  may  be  expelled.  Return  it  to  the  freezing 

1  Richardson,  Jour.  Chem.  Soc.,  1887,  ^>  P-  397- 


THE    PHASE    RULE.  267 

mixture,  refill  the  condensation  vessel,  and  again  condense  nitrogen 
peroxide  in  the  bulb. 

Finally,  disconnect  the  bulb,  heat  it  to  about  20°,  and  immediately 
connect  it  to  the  manometer  of  the  air  thermometer.  The  mercury 
should  be  at  such  a  level  that  there  is  little  air  space  between  it 
and  the  bulb. 

Surround  the  bulb  with  ice  and  water  and  make  a  number  of 
independent  readings  of  the  mercury  levels.  Since  the  volume  is 
to  be  kept  constant,  the  mercury  on  the  side  next  the  bulb  must 
always  be  brought  to  the  same  position.  This  should  be  as  high  as 
possible  to  reduce  the  air  space. 

Replace  the  ice  by  water  at  about  20°  and  adjust,  and  read  the 
mercury  levels.  So  continue,  'for  about  every  20°,  until  the  water 
boils,  when  several  observations  should  be  made.  (See  Experiment 
III.)  Read  the  barometer  and  calculate  the  pressures.  Plot 
pressures  against  temperatures  as  abscissae.  Find  the  temperature, 
t,  for  which  the  pressure  is  76  cm.,  and  from  Fig.  85  find  the  degree 
of  dissociation  corresponding  to  this  temperature  and  pressure. 
From  Equation  181  calculate  the  pressure  p'  for  this  temperature, 
if  there  were  no  dissociation.  Find  the  corresponding  undissociated 
pressure  p"  for  100°  by  Gay  Lussac's  law. 


P"        373 

Locate  these  two  points  on  the  plot  and  connect  them  by  a  dotted 
straight  line. 

For  five  temperatures,  calculate  the  degree  of  dissociation  from 
the  ordinates  of  the  two  curves  and  Equation  181,  and  plot  the 
values  obtained. 

From  one  of  the  first  and  one  of  the  last  values  of  a  and  the  cor- 
responding absolute  ^temperatures,  calculate  the  heat  of  formation 
of  NO2  from  N2O4  at  constant  volume.  Remember  that  Q  is  the 
heat  absorbed,  or  —  Q  is  the  heat  emitted  per  gram  molecule  of  N2O4, 
in  the  reaction 


QUESTIONS. 

1.  What  is  the  heat  of  formation  of  one  gram  of  NO2? 

2.  Is  the  formation  of  NO2  endothermic  or  exothermic? 

3.  From    Q,    and   the   degree    of   dissociation   at    some   particular 
temperature,   calculate  the  temperature   centigrade  at   which   99% 
of  the  gas  is  dissociated. 

The    Phase    Rule.1 

Thus  far  we  have  considered  the  relations  between  the  con- 
centrations of  different  molecules  in  one  state,  and  of  the 
same  molecule  in  different  states.  We  have  also  considered 

1  Gibbs,  Trans.  Conn.  Acad.,  1874-1878,11!,  pp.  108-303;  Collected  Memoirs, 
I,  p.  96;  Findlay,  The  Phase  Rule;  Bancroft,  The  Phase  Rule;  Juptner,  ii, 
Chap.  VII;  Whetham,  Chapters  II,  III. 


268  CHEMICAL    STATICS. 

how  the  concentrations  must  change  if  the  temperature  and 
pressure  (or  volume)  change.  But  we  have  not  learned  in 
how  many  different  states  a  molecule  may  exist  in  equilibrium, 
nor  what  changes  a  system  may  surfer  without  losing  one  or 
more  states. 

270.  We  must  first  define  several  terms.      By  "phase"  we 
will  understand  a  physically  homogeneous  state;  for  example, 
in  the  system  consisting  of  water  and  salt,  the  solution,  the 
vapor,   and   the   undissolved   salt"  at   the   bottom   are   three 
distinct  phases.     If  some  ice  were  present,  there  would  be  a 
fourth    phase.      By   "components'1   we   will    understand   the 
substances  composing  the  system.     In  the  above  illustration, 
the  components  are  salt  and  water.     As  the  phases  are  physical 
divisions,  so  the  components  are  chemical  divisions.     If  in 
any   case   there   is  uncertainty,  as  to  what,   or  how  many, 
components  should  be  chosen,   we  shall  always  take  those 
which  are  indispensable  in  describing  each  phase,  and  whose 
composition  in  each  phase  may  be  independently  varied.     In 
the  above  example  we  take  salt  as  one  component  rather  than 
sodium  and  chlorine,  because  the  concentrations  of  the  latter 
cannot  be  varied  independently. 

271.  In  some  cases  the  particular  choice  of  components  is 
uncertain  and  unimportant,  for  example,  in  the  equilibrium 
between  calcium  carbonate,  calcium  oxide  and  carbon  dioxide, 

CaCO3^±CaO+CO2 

any  two  of  the  three  substances  may  be  chosen,  for  two  are 
necessary  and  sufficient  to  chemically  describe  each  phase 
and  may  exist  in  independent  proportions. 

We  saw  in  §235  that  the  state  of  a  system  may  be  defined 
by  the  intensity  factors,  temperature,  pressure,  and  poten- 
tial of  each  component,  and  that  the  potential  in  a  particular 
phase  is  proportional  to  the  concentration.  It  was  also 
pointed  out  that  the  intensity  factors  are  the  same  throughout 
the  system,  and  do  not  depend  upon  the  actual  amount  of  any 
phase  or  component. 

272.  Let  P  be  the  number  of  phases  in  a  certain  system. 


THE    PHASE    RULE.  269 

Let  N  be  the  number  of  components.  There  are  therefore 
N-\-2  intensity  factors  in  each  phase  (N  potentials  and  pres- 
sure and  temperature).  How  many  of  these  can  be  varied 
without  disturbing  the  number  of  phases? 

The  mass  law  gives  us  one  necessary  relation  between  the 
concentrations  (and  therefore  potentials)  in  each  phase. 
There  remain  N  —  i  possible  variations  in  the  concentrations 
in  each  phase,  or,  with  temperature  and  pressure,  a  total 
number  of  independent  variables 

(N-i)  P  +  2 

for  the  entire  system.  The  potential  of  a  component  must, 
at  equilibrium,  be  the  same  in  every  phase.  To  express  this 
equality  mathematically  requires  N  equations  between  any 
particular  phase  and  the  other  P  —  i  phases,  or  a  total  of  N 
(P  —  i)  equations. 

The  number  of  undetermined  variables,  F,  is  therefore  equal 
to  the  number  of  variables,  less  the  number  of  equations,  or 

F=(N-i)P  +  2-N(P-i)  =  N  +  2-P  (183) 

This  is  the  famous  Phase  Rule  of  Willard  Gibbs.  In  words, 
it  states  that  the  number  of  possible  variations  of.  concentra*- 
tion,  pressure,  and  temperature,  without  changing  (reducing) 
the  number  of  phases,  is  two  more  than  the  difference  between 
the  number  of  components  and  the  number  of  phases.  Con- 
versely, if  we  determine  the  maximum  number  of  possible 
variations,  we  can  judge  of  the  number  of  phases  present. 

273.  By  variations  are  understood  arbitrary,  independent, 
assignments  of  particular  values  to  intensity  factors.  If  a 
particular  variation  involves  a  change  in  one  or  more  other 
factors,  the  two  or  more  together  constitute  but  one  variation. 
The  variations  of  concentration  must  be  variations  of  the 
concentration  of  one  or  more  phases,  not  simply  variations  in 
the  total  concentration.  For  concentration  is  used  as  a 
convenient  measure  of  potential,  and  the  total  concentration 
may  vary  on  account  of  a  change  in  the  relative  amounts  of  the 
different  phases  which  would  not  affect  the  concentration 
(and  potential)  in  each  phase. 


270 


CHEMICAL    STATICS. 


274.  The  phase  rule  is  of  particular  utility  in  conjunction 
with  a  graphical  representation  of  the  system,  with  the  inten- 
sity factors  as  coordinates.     If  there  is  one  component,   the 
axes  are  temperature  and  pressure,  and  the  different  phases 
are  represented  by  areas,  the  coexistence  of  two  phases  by 
lines,  and  of  three  phases  by  points. 

275.  Illustrations. — One  component.     F=$—P. 

Water.1 — The  different  phases  are  represented  by  the  areas 
designated   in   Fig.    86.     Two   phases   are   coexistent   at   the 


p  (cm.) 
i. 


E 

Solid 


•ID* 


Liquid 


0° 
FIG.  86. 


!0C 


temperatures  and  pressures  corresponding  to  the  lines  OA, 
OB  and  OC,  and  at  0,  where  the  temperature  is  .0076°  and 
the  pressure  is  4.6  mm.  (§154),  three  phases  coexist. 

At  a  point  E,  when  there  is  but  one  phase,  F  =  2,  or  the 
system  is  bivariant;  that  is,  we  may  independently  change 
both  temperature  and  pressure,  as  is  evident  from  the  figure. 
At  a  point  on  one  of  the  lines,  F  =  i,  or  the  system  is  uni- 
variant,  that  is,  we  may  arbitrarily  change  either  the  pressure 
or  the  temperature,  but  not  both,  if  the  two  phases  are  to  be 
preserved.  For  if  we  change  the  temperature,  there  must  be 

1  Jiiptner,  ii,  pp.  204-5;  Whetham,  pp.  39-44. 


ALLOYS. 


271 


an  accompanying  change  in  pressure  which  is  not  arbitrary, 
but  is  such  that  it  and  the  new  temperature  give  a  point  on 
that  line.  At  O,  F=o,  or  the  system  is  invariant.  The  dia- 
gram shows  that  at  this  point,  any  change  in  the  temperature 
or  pressure  will  reduce  the  number  of  phases. 

276.  Two  Components. — F=4—P.  Since  there  are  three 
intensity  factors,  temperature,  pressure,  and  concentration, 
a  figure  in  three  dimensions  is  required.  We  shall  usually 
find  it  most  convenient  to  lay  off  the  temperature  and  concen- 
tration along  horizontal  rectangular  axes,  in  which  case  the 
pressure  axis  is  vertical.  The  conditions  for  equilibrium  of 


1000° 


500° 


Ag+  Eutectic 


Cu  -*-  Eutectic 


1000° 


Ag 


Cone 
FIG.  87. 


Cu 


500 


vapor  and  solid,  or  vapor  and  solution,  or  vapor  and  both, 
will  lie  upon  a  surface.  Above  the  surface  the  pressure  is  so 
great  that  the  vapor  phase  is  absent,  below  the  surface  is 
only  the  vapor  phase. 

277.  Alloys.  Silver  Plus  Copper.1 — Fig.  87  represents  the 
surface  viewed  from  above,  that  is,  along  the  pressure  axis. 
Consider  the  point  D  on  this  surface.  Here  there  are  two 
phases,  solution  and  vapor,  and  therefore  there  are  two  possible 

1  Whetham,  pp.  59-63. 


272  CHEMICAL    STATICS. 

variations.  We  shall  choose  to  lower  the  temperature  and  keep 
the  concentration  the  same.  The  pressure  will  therefore 
change  in  a  definite  manner,  out  of  our  control,  if  we  wish  to 
preserve  the  vapor  phase.  (The  exact  mode  of  variation  of 
the  pressure  would  be  shown  by  a  section  of  the  surface, 
perpendicular  to  the  c  —t  plane,  and  through  the  line  DE.)  At 
E  solid  silver  separates  out,  there  are  three  phases,  and  the 
system  is  univariant.  If  we  lower  the  temperature,  the 
concentration  of  the  solution  is  beyond  our  control  and  varies 
with  the  temperature  to  give  a  line  such  as  EO.  At  a  point 
such  as  F,  the  total  concentration  has  not  changed,  but  the 
concentration  of  the  liquid  phase  has  changed  as  just  described, 
and  it  is  such  variations  of  concentration  which  are  contem- 
plated in  the  phase  rule  and  not  variations  in  the  total  con- 
centration (§273).  The  concentration  at  F  is  given  by  an 
amount  of  pure  silver  proportional  to  FG  and  of  solution 
proportional  to  HF.  The  solution  has  the  proportions, 
Ag:  Cu=GI:  GH 

At  O  solid  copper  begins  to  appear  from  the  solution,  and 
four  phases  are  present — vapor,  solution,  silver,  and  copper. 
The  system  at  this  point  is  therefore  invariant.  This  point  is 
called  the  eutectic  point,  and  the  alloy  which  separates  out  upon 
further  cooling,  of  the  concentration  of  the  point  O  [40  (atomic) 
parts  of  copper]  is  called  the  eutectic  alloy,  and  evidently 
has  a  definite  melting  point.  At  any  point  M  we  have  a 
mixture  (solid  solution,  §182)  of  solid  silver  and  solid  eutectic 
alloy  of  the  concentrations  indicated. 

Similar  diagrams  for  iron  and  carbon  are  somewhat  more 
complex,  but  are  of  great  industrial  importance.1 

278.  Dissociation  of  Calcium  Carbonate. — Upon  heating, 
calcium  carbonate  dissociates  into  calcium  oxide  and  carbon 
dioxide  (§262).  We  shall  consider  a  vertical  section,  perpen- 
dicular to  the  concentration  axis,  of  such  a  figure  as  is  described 
in  §276.  The  axes  are  pressure  and  temperature,  and  the 

1  Roozeboom,  Journ.  Iron  and  Steel  Inst.,  1900,  ii,  p.  311;  Zeit.  Elektro- 
chem.,  1904,  x,  p.  489;  Juptner,  i,  pp.  135-138;  Roberts-Austin,  "Intro,  to 
Metallurgy,"  5th  ed.,  p.  802;  Findlay,  "Phase  Rule,"  pp.  224-228. 


DISSOCIATION    OF    CALCIUM    CARBONATE. 


273 


curve  A  B,  Fig.  88,  is  the  section  of  a  suface  which  separates  the 
carbonate  above  (high  pressures)  and  the  oxide  and  gas  below 
(low  pressures).     Along  A  B  three  phases  are  present,  two 
solids  and  gas,  and  therefore  the  system  is  uni variant.     We 
may  arbitrarily  change  the  pressure,  but  unless  the  tempera- 
te (cm) 
I50| 


100 


50 


CaCO 


500° 


600° 


700" 

FIG.  88. 


800° 


900° 


ture  is  also  changed  to  correspond  with  a  point  on  the  curve, 
one,  or  two  phases  will  be  lost. 

Other  sections,  parallel  to  this  one,  would  correspond  to  a 
greater  or  less  concentration  of  CO2  in  the  lower  part  of  the 
figure,  and  therefore  the  curves  would  be  somewhat  different. 

EXPERIMENT  XV. 

This  experiment  should  now  be  reviewed  and  the  results  inter- 
preted with  the  help  of  the  phase  rule. 

PROBLEMS  XX. 

1.  When  CuSO4'  sH2O  crystals  are  heated,  we  obtain  successively 
CuSO4'    3H2O,   CuSCV    H2O   and    the  anhydrous   salt.      Determine 
qualitatively  how  the  temperature  will  vary. 

2.  Prove  that  the  vapor  pressures  of  ice  and  water  are  equal  at 
-.0076°. 

3.  Could  two  of  the  above  copper  sulphate  salts  remain  in  excess, 
in  equilibrium,  in  an  aqueous  solution? 

18 


274 


CHEMICAL    STATICS. 


4.  When    hydrogen    is    forced    into    paladium,    within    a    certain 
range,  the  volume  absorbed  is  proportional  to  the  pressure.      Upon 
increasing  the  pressure  further,   a  point   is  reached  where   a   very 
slight  increase  in  pressure  is  accompanied  by  a  great  increase  in  the 
volume  absorbed.      Decide  by  the  phase  rule  how  many  phases  are 
present  in  each  stage. 

5.  Decide  by  the  phase  rule  whether  the  eutectic  alloy  is  a  com- 
pound or  a  mixture. 

EXPERIMENT  XLII. 

Construction  of  a  Model  to  Represent  the  Equilibrium  of  Salt  and 

Water.' 

Upon  a  board  or  other  smooth  surface  mould  a  block  of  modeling 
clay  or  plaster  of  Paris,  making  the  base  about   10  cm.  by   15  cm. 


NaCI. 


H20 


Salt 

+ 

Cryo- 
hydrate 


Cryo-  j 
hydrate; 


Salt 

+- 

Saturated 
So.lution 

W 


-30        A -20 


-10 
temp  — 

FIG.  89. 


10 


20 


While  the  clay  or  plaster  is  sufficiently  soft  to  cut  easily,  form  the 
surface  which  will  show  the  equilibrium  between  the  vapor  and  other 
phases  within  the  limits  of  about  —30°  and  20°  and  o  and  1.5  cm. 

1  Van't  Hoff,  i,  pp.  32-38;  Findlay,  "Phase  Rule,"  pp.  126-132;  Juptner.  ii, 
pp.  225-227;  Whetham,  pp.  49-52.' 


CRYOHYDRATES.  275 

pressure.  Choose  one  of  the  bottom  corners  as  the  origin.  Make 
the  longer  horizontal  edge  the  axis  of  concentration,  the  other 
horizontal  edge  the  temperature  axis,  and  the  vertical  edge  the 
pressure  axis.  Fig.  89  represents  roughly  the  appearance  of  such 
a  surface  if  it  is  viewed  from  above. 

The  face  inclosed  by  the  temperature  and  pressure  axes  should 
appear  like  the  upper  curve  of  Fig.  71,  since  the  concentration  of  the 
salt  is  zero.  The  surface  will  slope  away  from  this  curve  since  the 
greater  the  amount  of  salt,  the  less  the  pressure.  The  boundaries 
between  different  phases,  represented  by  the  lines  OA,  OB,  OC,  OD, 
OE,  will  appear  as  more  or  less  abrupt  changes  in  slope. 

Below  the  surface  the  pressure  is  so  low  that  there  is  only  vapor, 
while  above  the  surface  the  vapor  phase  is  absent. 

A  point  P  in  such  a  region  on  the  surface  as  OAD  represents  the 
coexistence  of  three  phases — vapor,  unsaturated  solution,  and  ice. 
The  amounts  of  the  two  latter  are  in  the  proportion  PN :  MP.  The 
point  O,  where  CO  and  OD  meet,  represents  the  coexistence  of  salt, 
saturated  solution,  ice,  and  vapor.  Since  there  are  four  phases,  the 
system  is  invariant.  The  temperature  is  —22°,  the  concentration 
is  23.5%,  and  the  pressure  is  .079  cm.  This  point  is  called  the 
cryohydrate  point.  If  the  system  is  cooled  from  the  point  O,  salt 
and  ice  solidify  in  the  above  proportions,  and  the  mixture  is  called 
the  cryohydrate.  A  point  such  as  Q  represents  a  mixture  of  ice  and 
cryohydrate  in  the  proportions  QT  :  QS.  OC  is  evidently  the 
solubility  curve  and  OD  is  the  freezing-point  curve. 

The  lower  pressures  are  so  small  that  they  should  be  exaggerated 
in  the  model.  Designate  the  phases  present  in  the  different  portions 
of  the  surface. 


QUESTIONS. 

1.  What  is  the  ratio  of  the  concentration  of,    (a)   salt  and  cryo- 
hydrate, (b)  salt  and  water,  corresponding  to  a  point  such  as  X? 

2.  (a)   What  are  the  relative  concentrations  of  salt  and  saturated 
solution  corresponding  to  a  point  such  as  Wf     (b)  of  salt  and  water? 

3.  Explain  how  and  where  the  phases  change  when  a  mixture  of 
salt  and  water  is  cooled  from    +10  to    —30°,  if  the  initial  state  cor- 
responds to  (a)  the  point  V,  (b)  the  point  U. 

4.  What  are  the  phases  at  a  point  a  little  above  M?  below  M? 

EXPERIMENT  XLIII. 
The  Melting  Point  of  an  Alloy. 

If  an  alloy  is  melted  and  is  allowed  to  cool,  while  its  temperature 
is  continuously  observed  and  a  curve  be  then  drawn  with  times  as 
abscissae  and  temperatures  as  ordinates,  it  will  be  found  that  at 
certain  points  the  curvature  abruptly  changes,  the  fall  of  temperature 
being  decreased  or  even  ceasing.  At  the  moment  corresponding 
to  such  a  point,  the  alloy  is  radiating  heat  to  the  room,  and  the  fact 
that  its  temperature  does  not  fall  as  rapidly  indicates  that  heat 
is  being  produced  internally  by  some  change  of  state  of  the  material. 
Such  a  point  is  therefore  a  solidifying  point  of  some  constituent 
of  the  alloy  or  of  the  eutectic  alloy  (§277). 


276  CHEMICAL    STATICS. 

The  assigned1  metals  are  carefully  weighed  and  melted  in  an  iron 
cup.  A  copper-constantin  thermo  couple  (§49)  is  plunged  into 
the  liquid  metal  and  kept  there  until  the  rentie  mass  is  solid.  A 
porcelain  tube  2  should  cover  one  wire  for  some  distance  from  the 
junction.  The  terminals  are  connected  to  a  galvanometer  through 
a  resistance  such  that  the  maximum  deflection  will  keep  on  the  scale. 
The  galvanometer  is  read  every  half  minute  and  the  time  of  each 
reading  is  noted.  When  the  readings  are  commenced,  the  metal 
should  be  considerably  above  the  melting  point  and  the  readings 
should  be  continued  for  some  time  after  the  metal  is  apparently  solid. 

Find  the  number  of  volts  or  millivolts  corresponding  to  unit 
deflection  of  the  galvanometer  by  applying  a  known  small  electro- 
motive force  to  the  galvanometer,  with  its  resistance,  and  observing  the 
deflection.  (§§47,77.)  Plot  the  galvanometer  deflections  against  the 
time.  Determine  the  electromotive  forces  corresponding  to  the  gal- 
vanometer deflections  where  the  curvature  changed,  and  from  the  con- 
stants of  the  thermocouple,  or  a  chart  giving  the  temperature  for  dif- 
ferent electromotive  forces,  determine  the  temperatures  of  these  points. 

Tabulate  the  observed  temperatures  of  these  transition  points 
and  your  opinion  of  what  they  represent. 

QUESTIONS. 

1.  Explain   why   the    second   transition   point   is   represented   by 
a  horizontal  portion  of  the  cooling  curve,  .while  the  first  transition 
point  is  merely  represented  by  a  change  of  curvature. 

2.  Will  the  temperature  of  the  first  point  vary  with  the  initial 
concentration? 

3.  Will  the  temperature  of  the  second  transition  point  vary  with 
the  initial  concentration? 

279.  Transition  Points. — The  points  where  several  phases 
coexist  are  of  considerable  importance.  We  have  already 
considered  the  eutectic  alloy  and  the  cry ohy drat e.  Other 
examples  will  be  found  in  special  treatises. 

In  some  cases  it  is  difficult  to  observe  such  a  point  directly, 
but  we  can  find  it  by  slightly  varying  the  temperature  or 
pressure,  and  observing  the  effect  on  the  volume,  temperature, 
or  pressure  caused  by  the  disappearance  of  one  phase,  or  the 
appearance  of  another,  or  both  effects.  These  points  of  coex- 
istence of  several  phases  are  often  called  transition  points. 

1  Tin  and  lead  are  suitable  metals.     The  changes  of  curvature  are  more  dis- 
tinct if  the  former  is  in  excess.     It  is  interesting  to  have  different  students  use 
different  alloys  and  compare  the  results  when  all  have  completed  the  experi- 
ment.    The  eutectic  of  tin  and  lead  is  composed  of  37%  lead,  63%  tin,  and 
melts  at  182.5°  (Rosenhain  and  Tucker,  Roy.  Soc.  Phil.  Trans.,  1908,  A.  209, 
p.  89).     For  further  interesting  information  regarding  this  alloy  see  Zeit.  elect. 
Chemie,  1909,  xv,  p.  125. 

2  Fine  porcelain  tubes  may  be  obtained  of  the  Royal  Berlin  Porcelain  Factory 
or  American  Agents. 


TRANSITION    POINT    OF    SODIUM    SULPHATE.  277 

280.  Transition  Point  of  Sodium  Sulphate.1 — If  we  have  a 
saturated  solution  of  sodium  decahydrate,  together  with  an 
excess  of  hydrated  salt,  and  heat  it  gradually,  at  32.384°, 2  the 
hydrate  changes  to  the  anhydrous  salt  plus  water,  and  the 
solution  of  the  anhydrous  salt  has  a  slightly  larger  volume  than 
the  solution  of  the  hydrate.  Above  this  temperature  the 
anhydrous  salt  and  its  solution  are  stable,  if  the  pressure  is 
suitable.  At  the  above  temperature,  the  two  salts,  their 
common  solution,  and  vapor  coexist,  and  this  is  the  transition 
point. 

EXPERIMENT  XLIV. 
Transition  Point  of  Sodium  Sulphate. 

Prepare  a  dilatometer  as  follows:  Procure  a  "weight  thermom- 
eter "  ;  that  is,  a  glass  bulb  with  a  long  capillary  stem.  The  capacity 
of  the  bulb  should  be  about  5  c.c.  and  the  stem  should  have 
an  internal  diameter  of  about  .7  mm.  and  a  length  of  about 
2  5  cm.  By  means  of  a  one-hole  rubber  stopper  and  a  large 
glass  tube,  provide  a  reservoir  at  the  top  of  the  capillary 
and  fill  it  with  crystals  of  sodium  decahydrate,  together  with 
a  very  little  water.  (See  Fig.  90.) 

Warm  very  gently  the  bulb,  stem,  and  reservoir.  The 
hydrate  will  be  melted  and  the  warmed  air  will  bubble  out 
through  the  liquid.  Be  careful  not  to  heat  the  stem  and 
reservoir  too  hot,  otherwise  considerable  anhydrous  salt  will 
separate  out.  When  some  of  the  melted  salt  has  been  col- 
lected in  the  bottom  of  the  bulb,  it  may  be  heated  until  all 
of  the  air  has  been  expelled  by  the  vapor.  Now  cool  gradu- 
ally. When  the  bulb  is  nearly  full,  replace  the  remaining 
liquid  in  the  reservoir  with  kerosene,  which  will  prevent 
evaporation  of  the  liquid  below  and  serve  as  an  index  in  the 
capillary.  Adjust  by  suitable  heating  and  cooling  until  the 
level  of  the  kerosene  is  in  the  lower  half  of  the  capillary. 

Remove  the  reservoir  and  attach  a  paper  scale.  Place 
the  bulb  in  a  vessel  of  water  provided  with  a  1/10°  ther- 
mometer. Gradually  raise  the  temperature  of  the  bath  and 
read  the  level  of  the  meniscus  at  every  half  degree  between 
30°  and  32°,  every  fifth  degree  between  32°  and  35°,  and 
every  half  degree  between  35°  and  37°.  Repeat  with  de- 
scending temperatures. 

Plot  your  results  with  temperatures  as  abscissae  and  posi-     ^—^ 
tions  of  the   meniscus    (volumes)   as  ordinates.      The  two    pIG   ^0. 
curves  will  show  a  lag.     Explain.     State  the  mean  value  of 
the  temperature  of  the  transition  point. 

Another  method  which  should  be  employed,  if  time  permit,  is  to 
place  the  bulb  in  a  thermostat  at  about  32°  and  see  if  there  is  any 

1  Findlay,  "Phase  Rule,"  pp.   134-142;  Whetham,  pp.   53-56;  Juptner,  ii, 
pp.  233-235. 

2  Dickinson  and  Mueller,  Bui.  Bu.  Stand.,  1907,  iv,  p.  641. 


278  CHEMICAL    STATICS. 

change  of  volume,  and  then  successively  at  32.5°,  33°,  33.5°,  34°,  etc. 
There  is  likely  to  be  some  of  the  unstable  salt  present  and  therefore 
there  will  be  a  continuous  change  of  volume  except  at  the  transition 
temperature. 

This  experiment  may  be  varied  by  substituting  for  the  sodium 
decahydrate  an  approximately  equimolecular  mixture  of  sodium 
decahydrate  and  magnesium  sulphate  heptahydrate.  At  about  21°, 
astrachanite,  Na2Mg(SO4)2'  4H2O,  and  water  separate  out  and  the 
transition  point  can  be  determined  with  a  dilatometer. 

For  the  solubility,  thermometric,  and  tensimetric  methods  of  de- 
termining the  transition  temperature,  see  special  treatises.1 

QUESTIONS. 

1.  What  other  invariant  point  has  sodium  sulphate  and  water? 

2.  Explain  which  dilatometer  method  you  prefer. 

iFindlay,  "Prac.  Phys.  Chem,"  Chap.  XV;  "Phase  Rule,"  Appendix; 
Juptner,  ii,  pp.  205-207;  Nernst,  pp.  588-589. 


CHAPTER  VIII. 
ELECTROCHEMISTRY,1  ELECTROLYTIC  CONDUCTION. 

281.  Faraday's  Law. — The  greatest  contributor  to  our 
knowledge  of  this  subject  was  Michael  Faraday  (1791-1867). 
The  most  important  law  which  he  discovered  will  be  stated  in 
his  own  words  :2  "  The  chemical  decomposing  action  of  a  current 
is  constant  for  a  constant  quantity  of  electricity,  notwithstanding 
the  greatest  variation  in  its  sources,  in  its  intensity,  in  the 
size  of  the  conductors  used,  in  the  nature  of  the  conductors 
(or  non-conductors)  through  which  it  is  passed,  or  in  other 
circumstances."  Faraday  also  invented  our  present  termin- 
ology. "I  propose  to  call  bodies  of  this  the  decomposable 
class,  electrolytes.  The  anode  is  ...  that  surface  at  which 
the  electric  current  enters  the  electrolyte,  it  ...  is  where 
oxygen,  chlorine,  .  .  .  etc.,  are  evolved.  .  .  .  The  cathode 
is  that  surface  at  which  the  current  leaves  the  decomposing 
body  .  .  .  metals,  alkalies  .  .  .  are  evolved  there."  "Then, 
again,  the  substances  into  which  these  divide  under  the 
influence  of  the  electric  current  form  an  exceedingly  important 
general  class.  They  are  combining  bodies,  are  directly  asso- 
ciated with  the  fundamental  parts  of  the  doctrine  of  chemical 
affinity,  and  have  each  a  definite  proportion  in  which  they 
are  always  evolved  during  electrolytic  action .  I  have  proposed 
to  call  these  bodies  generally  ions,  or  particularly  anions 
or  cations,  according  as  they  appear  at  the  anode  or  cathode, 
and  the  numbers  representing  the  proportions  in  which  they 
are  evolved  electrochemical  equivalents.  Thus  hydrogen, 

1  General   references    "Electrochemistry,"    Lehfeld,    Longmans;     "Electro- 
chemistry," Arrhenius,  Longmans;   "Electrochemistry,"  LeBlanc,  Macmillan; 
"Electro-chemie,"    van    Laar,    Engelmann,    Leipsic    (founded    upon    thermo- 
dynamic  potentials);  Winkelmann,  1905,  iv,  I. 

2  Experimental   Researches,  vol.  i,    ser.  Ill,  VII;    also   in   "Fund.  Laws  of 
Electrolytic  Cond.,"  Goodwin. 


280       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

oxygen,  chlorine,   iodine,   lead,  tin,   are  ions  .   .   .  and    i,  8, 
36,  125,  104,  58,  are  their  electrochemical  equivalents." 

"Electrochemical  equivalents  coincide,  and  are  the  same, 
with  ordinary  chemical  equivalents." 

282.  To  avoid  this  unnecessary  duplication  of  names,  the 
term  electrochemical  equivalent  is  now  applied  to  a  fraction,  one 
ninety-six  thousand  five  hundred    and    thirtieth  of   the  above 
numbers.     This  fraction  is  the  mass  of  hydrogen,  in  grams, 
liberated  by  one  coulomb  (§io).x     Therefore  96,530  coulombs 
(which,   for  convenience,   is   often   called   one  faraday)    will 
liberate  one  gram  of  hydrogen,  eight  grams  of  oxygen,   35,5 
grams  of  chlorine,  or,  in  general,  a  mass  numerically  equal  to 
the  chemical  equivalent. 

The  chemical  equivalent  is  the  atomic  (or  ionic)  weight 
divided  by  the  valency  and  the  electrochemical  equivalent  is 
the  chemical  equivalent,  divided  by  96,530.  Table  L  gives 
the  atomic  weights  of  the  more  common  elements.  The 
corresponding  weight  of  a  radical  ion  is  the  sum  of  the  atomic 
weights  of  the  constituents.  The  valencies  are  so  familiar 
that  they  are  not  tabulated. 

283.  Faraday's  description  (see  reference)  of  the  experiments 
by  which  he  substantiated  his  law  is  very  interesting  and  in- 
structive.    Since  Faraday's  time  the  law  has  been  subjected  to 
more   exact   experimental   tests.     The   discrepancies   do  not 
exceed  the  possible  experimental  errors.     Perhaps  the  most 
accurate  work  is  that  of   Richards  and  Stull2  who  sent  the 
same  current  through  an  aqueous  solution  of  silver  nitrate 
at   20°  and  a  solution   of    silver  nitrate  in  melted    sodium 
potassium  nitrate  at  260°.     The  mean  difference  in  the  mass 
of  silver  liberated  in  the  two  solvents  was  .005%.     Kahlen- 
berg3  compared    the  deposition  of   silver   from    an    aqueous 
solution  of  silver  nitrate  with  that  from  various  organic  sol- 
vents, such  as  pyridine,  aniline,  etc.,  and  found  the  differences 
less   than    the   experimental   errors.     Richards,  Collins,  and 

1  Van  Dijk,  Arch.  Neerl.,  1905,  10,  287. 
2Zeit.  phys.  Chem.,  1903,  xlii,  p.  621. 
3  Jour.  Phys.  Chem.,  1900,  iv,  p.  349. 


DISSOCIATION    THEORY.  281 

Heimrod1  found  the  ratio  of  the  masses  of  silver  and 
copper  deposited  in  two  solutions,  traversed  by  the  same  cur- 
rent, to  be  3.3940,  while  the  ratio  of  their  chemical  equivalents 
was  3. 3938. 

284.  Dissociation  Theory. — Thus  far  we  have  only  considered 
the  liberation  or,  as  Faraday  termed  it,  the  decomposition, 
at    the  electrodes.     Arrhenius  suggested,  in  i88y,2  a  theory 
which  is  now  generally  accepted,  namely,  that  the  substances 
which  were  evolved  at  the  electrodes  existed  in  the  solutions 
as  independent  ions  before  the  electric  current  passed  through, 
and  before  the  solution  was  subjected  to  any  electric  forces. 
For  example,  according  to  the  theory  of  Arrhenius,  when  silver 
nitrate  is  dissolved  in  water,   a  certain  portion  remains  as 
molecules  of  silver  nitrate,   but  the  remainder   (usually  the 
larger  portion)  breaks  up  into  its  constituents,  silver  and  the 
nitrate  radical,  with  equal  opposite  electric  charges. 

285.  Electric   Charge   on    Ions. — We    understand    by   the 
direction  of  a  current,  the  direction  in  which  positive  electricity 
moves,  and  since  the  cations  follow  the  current,  they  must 
carry  positive  charges.     Hydrogen  and  all  metallic  ions  are 
cations. 

A  solution  has  no  free  charge  of  electricity,  and  therefore 
the  sum  of  the  positive  charges  on  the  cations  must  equal  the 
sum  of  the  negative  charges  on  the  anions. 

As  required  by  Faraday's  law,  each  ion  has  always  a  definite 
charge  in  whatever  combination  it  may  occur.  Therefore, 
since  a  cation  such  as  sodium  occurs  singly  with  such  an  anion 
as  chlorine  and  twice  with  the  sulphate  radical,  the  charges  on 
the  two  anions  cannot  be  the  same,  but  must  be  proportional  to 
the  valencies .  Evidently  a  similar  law  must  be  true  for  cations . 
Therefore,  the  charge  on  a  copper  ion  is  twice  that  on  a  silver 
ion  and  equal  to  that  on  a  ferrous  ion.  -  The  charge  on  a 
ferric  ion  is  three  times  that  on  a  silver  ion. 

The  kinetic  theory  of  gases  enables  us  to  estimate  the 
approximate  number  of  ions  in  a  given  mass,  and  thus  we  can 

JZeit.  phys.  Chem.,  1900,  xxxii,  p.  321;  1902,  xli,  p.  302. 
2Zeit.  phys.  Chem.,  1887,  i,  p.  631. 


282   ELECTROCHEMISTRY,  ELECTROLYTIC  CONDUCTION. 

determine  the  order  of  the  charge  on  a  single  ion.  The 
number  of  molecules  in  a  cubic  centimeter  of  gas,  under  standard 
conditions  (§4),  is  probably  of  the  order  ioI9.(x)  96,530 
coulombs  liberate,  and  are  therefore  carried  by,  one  gram  of 
hydrogen,  which,  under  standard  conditions,  occupies  11,120 
c.c.  The  number  of  ions  will  equal  the  number  of  atoms  or 
twice  the  number  of  molecules.  The  charge  on  a  univalent 
ion  is  therefore  of  the  order 

9  53°      _  =  4xio-19  coulombs 

1120X2  Xio'9 

286.  Voltameter. — Faraday's  law  suggests  a  simple  and 
accurate  method  for  determining  the  quantity  of  electricity 
which  traverses  a  circuit. 

A  suitable  electrolyte  is  made  a  part  of  the  circuit  and  the 
mass  liberated  at  one  of  the  electrodes  is  determined.  The 
quotient  of  this  mass  divided  by  the  electrochemical  equiva- 
lent is  obviously  the  quantity  of  electricity  in  coulombs.  Such 
an  instrument  is  called  a  voltameter  or  coulometer. 

The  electrochemical  equivalent  of  silver  has  been  deter- 
mined with  the  greatest  care  and  the  mean  result  in  grams  per 
coulomb  is  .oouiyS.2  Although  the  silver  voltameter  is  the 
more  accurate,  a  voltameter  where  copper  is  liberated  from 
copper  sulphate  is  more  convenient  (see  §79).  The  mean  cur- 
rent, in  amperes,  is  the  quantity,  in  coulombs,  divided  by 
the  time  (§10). 

PROBLEMS  XXI. 

1.  A  current  of  5  amperes  flows  for  one  hour  through  a  copper 
voltmeter,      (a)     How    many    coulombs    traverse    the    electrolyte? 
(6)   How  much  copper  is  deposited?      (c)   How  much  copper  is  dis- 
solved from  the  anode  ? 

2.  If   10   volts  are  applied  to  the  electrodes  of  a   certain  silver 
plating  bath,    20   grams  are   deposited  in  half  an   hour.      Find    (a) 
number  of  coulombs;    (6)    average   current;   (c)    average  power  ex- 
pended; (d)  work  done  by  current. 

3.  Five  Edison  cells  ( §340)  in  series  send  a  current  of  10  amperes  for 
one  hour  through  a  gold-plating  bath.     Find  (a)  total  zinc  dissolved 

IMeyer,  "Kin.  Theory  of  Gases,"  §120;  Boynton,  "Kin.  Theory,"  pp.  278-280. 
2Guthe,  Bui.  Bureau  of  Standards,  1905,  i,  3,  p.  362. 


COPPER    VOLTAMETER. 


283 


in  cells;  (6)  gold  deposited.  If  the  electric  motive  force  of  each  cell 
is  .9  volts,  calculate  (c)  the  average  power  and  (d)  the  total  work 
done  (§10). 

4.  Calculate  the  approximate  number  of  ions  in   i   c.c.   of  a  .01 
normal   solution   of  sodium   chloride   if   92%    of  the   molecules  are 
dissociated. 

5.  Calculate    the    electrochemical    equivalents    of     (a)    zinc;    (6) 
ferrous  iron;  (c)  ferric  iron;  (d)  SO4;  (e)  NO3. 


EXPERIMENT  XLV. 
Copper  Voltameter,  Joule's  Law. 

This  experiment  is  an  application  of  the  laws  of  Faraday  and 
Joule  and  it  affords  practice  in  the  use  of  a  copper  voltmeter  and  in 
interchanging  electrical,  thermal,  and  mechanical  units. 

A  current  is  sent  for  a  definite  time,  T,  through  a  copper  volt- 
meter and  through  a  coil  of  resistance,  R,  immersed  in  mw  gr.  of 
water  in  a  calorimeter  whose  total  water  equivalent  is  mc.  If  the 
mass  of  copper  deposited  is  m,  the  quantity  of  electricity,  e  (in 
coulombs),  which  has  traversed  the  circuit  is  m  divided  by  the 
electrochemical  equivalent  of  copper,  or  .0003292,  and  the  mean 
current,  i,  is  this  quotient  divided  by  the  time,  T.  The  work  done 
in  the  calorimeter,  W,  is  e  multiplied  by  the  fall  of  potential  in  volts 
and  the  latter,  by  Ohm's  law,  is  the  product  of  the  resistance  in 
Ohms  multiplied  by  the  current  in  amperes,  or  Ri. 

.'.  W  =  eiR  =i2RT  joules  (Joule's  law,  §10) 

The  heat  produced  in  the  calorimeter  (in  calories)  is  (mw+mc) 
(t2—tl)  where  tt  and  t2  are  the  initial  and  final  corrected  tempera- 
tures. 

If  ]'  represents  the  number  of  joules  in  one  calorie 


eiR=J' 


(184) 


The  mechanical  equivalent  of  heat,  usually  represented  by  Jt  is 
the  number  of  ergs  equivalent  to  one  calorie,  or  /  =  10?  ]' . 

Fig.  91  illustrates  the  connections. 

The  construction  and  operation  of  the  copper  voltameter  is 
fully  described  in  §79  of  the  introduction.  The  heating  coil  and 
calorimeter  are  shown  on  the  left.  The  latter  should  consist  of  a 
thin-walled,  brightly  polished  metal  vessel  of  about  200  c.c.  capacity, 
which  is  supported  on  corks  inside  a  larger  vessel,  the  space  between 
the  two  being  filled  with  cotton  wool.  The  heating  coil  should 
consists  of  a  spiral  of  constantin  or  other  resistance  wire  of  low- 
temperature  coefficient.  The  ends  of  the  coil  should  be  attached 
to  heavy  leads,  which  pass  through  a  wooden  top,  which  covers  both 
calorimeter  vessels,  and  which  has  a  hole  in  the  center  for  a  i  /io° 
thermometer.  The  resistance  of  the  coil  should  be  determined  with 
great  care  (§§65-67).  Between  i  and  2  ohms  is  a  convenient 
value.  The  inner  calorimeter  vessel  should  be  filled  with  distilled 
water  from  a  burette  to  such  a  height  that  the  coil  will  be  well 
covered.  The  water  should  preferably  be  several  degrees  below 
the  room  temperature.  The  storage  battery  supply  is  represented  at 


284       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

the  bottom  of  the  figure;  a  regulating  rheostat  on  the  left,  and  an 
ammeter  or  tangent  galvanometer  at  the  top.  If  the  latter  is  used, 
connections  must  be  made  through  a  reversing  switch.  Some  such 
instrument  is  needed  to  obtain  a  rough  idea  of  the  magnitude  of 
the  current  and  to  see  that  it  remains  constant.  The  current  should 
be  such  as  will  give  a  rise  of  10°  to  15°  in  half  an  hour.  When 
everything  is  ready,  read  the  temperature,  to  tenths  of  a  degree, 
every  minute  for  five  minutes,  and  then,  noticing  the  exact  minute, 


FIG.  91. 

close  the  switch.  Continue  minute  readings  of  the  temperature  and 
keep  the  current  steady  by  adjustments  of  the  rheostat.  When 
the  temperature  has  risen  about  10  or  15  degrees,  note  the  exact 
time,  open  the  switch  and  take  five  more  readings  of  the  tempera- 
ture at  minute  intervals.  Correct  for  radiation  as  described  in  §59. 

Calculate  from  your  observations  and  the  above  equations,  a 
value  for  the  mechanical  equivalent  of  heat. 

The  true  value  in  terms  of  the  15°  calorie  is  4.187  X  io7.  (*) 

QUESTIONS. 

1.  Calculate  the  mean  voltage  at  the  terminals  of  the  heating  coil. 

2.  What  error  would  be  introduced  if  this  voltage  was  excessive 

(§333)? 

1  Ames,  Rap.  Paris  Cong.,  1900,  i,  p.  204. 


CONDUCTION  THROUGH  AN  ELECTROLYTE.       285 

3.  Why  was  it  necessary  to  keep  the  current  constant?      (Note 
that  the  heat  produced  is  proportional  to  i*.) 

4.  What  percentage  error  would  be  introduced  by  neglect  of  the 
radiation  correction? 

5.  If  a  tangent  galvanometer  has  been  used,  calculate  the  hori- 
zontal component   of  the  earth's  magnetic  field. 

287.  Conduction  Through  an  Electrolyte. — Let  us  examine 
more  minutely  the  process  by  which  electricity  is  transported 
through  an  electrolyte. 

For  definiteness,  we  shall  consider  a  solution  of  silver 
nitrate,  in  which  dip  two  silver  plates.  A  moderate  difference 
of  potential  is  applied  to  these  plates;  for  example,  by  connect- 
ing them  to  a  storage  cell.  The  positively  charged  plate, 
the  anode,  will  attract  the  negatively  charged  NO3  ions  and 
repel  the  positively  charged  Ag  ions,  and  a  current  of  elec- 
tricity will  result. 

We  consider  as  the  direction  of  an  electric  current,  the  direc- 
tion in  which  positive  electricity  moves,  but  the  magnitude 
of  the  current  is  the  sum  of  the  current  in  the  positive  direc- 
tion plus  the  current  of  negative  electricity  in  the  opposite 
direction. 

At  the  cathode  the  entire  current  is  carried  from  the 
solution  by  the  positively  charged  silver  ions  to  the  silver 
plate,  and  at  the  anode  an  equal  amount  of  positive  electricity 
is  carried  by  an  equal  mass  of  silver  ions  from  the  anode  silver 
plate  to  the  solution.  In  the  solution  the  total  current  is  the 
positive  current  carried  by  the  Ag  ions  plus  the  negative 
current,  in  the  opposite  direction,  carried  by  the  NO3  ions. 

As  the  silver  ions  are  deposited  on  the  cathode,  they  must 
change  from  the  ionic  to  the  neutral  metallic  condition.  A 
positive  univalent  ion  may  be  considered  as  a  neutral  atom 
or  radical  from  which  one  electron  (§345)  or  ultimate  particle 
of  negative  electricity  has  been  removed.  The  negative 
charge  on  the  cathode  restores  the  electron  and  we  have 

Ag'  +  0    =Ag 

(One  dot  signifies  an  ion  with  a  single  positive  ionic  charge, 
and  a  dash  signifies  an  ion  with  a  single,  negative,  ionic  charge. 


286   ELECTROCHEMISTRY,  ELECTROLYTIC  CONDUCTION. 

0  is  the  symbol  for  an  electron).  A  negative  univalent 
ion  may  similarly  be  considered  as  a  neutral  atom  or  radical 
plus  an  electron. 

288.  Velocity  of  Ions.  —  Consider  the  current  through  a 
centimeter  tube  of  an  electrolyte  which  dissociates  into  two 
univalent  ions.  If  the  concentration  is  n  gram  equivalents 
per  litre,  the  number  of  equivalents  in  this  cubic  centimeter  is 
n/  1000. 

If  the  fraction  a  is  dissociated,  the  ionic  charge  on  either 
positive  or  negative  ions  must  be  96530^071000  for  one  gram 
equivalent  of  ions  has  a  charge  of  96,530  coulombs. 

Let  HI  =  velocity  of  positive  ions,  u2=  velocity  of  negative 
ions,  i  =  total  current 


For,  the  current  is  the  charge  which  passes  through  one  face 
of  the  cube  per  second. 

Since  we  are  considering  a  centimeter  cube,  the  current 
is  also  equal  to  the  specific  conductivity,  k  (§10),  multiplied 
by  the  difference  of  potential.  E  (which  is  also  the  potential 
gradient,  since  the  distance  is  i  cm.) 

/.  i  ooo  &£=  965307*0:  (uI-\-u2) 

If  E  is  in  volts  and  k  in  reciprocal  ohms,  HI  and  u2  are  in  centi- 
meters per  second.  For,  to  change  to  absolute  electromagnetic 
units  (Table  III),  the  factor  for  k  is  10-9,  that  for  E  is  io8,  and  that 
for  96,530  is  io-1;  therefore  the  total  factor  is  unity. 

k 

.'.  1000  —  =  96,530  a  (£/!  +  U2)  =A  (J86) 

n 

where  U  \  and  U2  are  the  velocities  of  the  ions  for  unit  potential 
gradient  (i  volt  per  cm.). 

289.  Equivalent  and  Molecular  Conductivity.  —  1000  k/n 
is  called  the  equivalent  conductivity,  and  will  be  designated  by  A. 
The  reason  for  the  factor  1000  is  that  in  electrical  units  (such 
as  specific  conductivity,  specific  resistance,  etc.)  the  cubic 


DEGREE    OF    DISSOCIATION.  287 

centimeter  is  the  unit  rather  than  the  cubic  decimeter  or  litre 
and  n/  1000  is  the  concentration  in  gram  equivalents  per  cubic 
centimeter. 

The  molecular  conductivity  is,  similarly,  one  thousand  times 
the  specific  conductivity  divided  by  the  concentration  in 
gram  molecules  per  litre.  It  is  therefore  equal  to  the  equiva- 
lent conductivity  multiplied  by  the  valency. 

We  can  obtain  a  physical  idea  of  ^  if  we  imagine  one  gram 
equivalent  of  the  solution  contained  in  a  tank,  two  of  whose 
sides  are  parallel  conducting  plates  i  cm.  apart.  Since  the 
concentration  of  i  c.c.  is  n/iooo,  IOOO/H  c.c.  of  solution  must 
be  between  the  plates.  The  conductivity  of  one  cubic  centi- 
meter is  k  (§10)  and  the  total  conductivity  between  the  two 
plates  must  be 


or,  the  equivalent  conductivity  is  the  conductivity  of  sufficient 
solution  to  contain  one  gram  equivalent,  inclosed  between  two 
parallel  plates  i  cm.  apart. 

290.  Degree  of  Dissociation.  —  It  is  found  by  experiment  that 
if  we  add  distilled  water  to  such  a  solution,  in  such  a  tank, 
the  conductivity  between  the  conducting  sides,  and  therefore 
the  equivalent  conductivity,  increases.  Therefore,  by  Equa- 
tion 1  86,  dilution  must  increase  a  or  increase  £/x  and  U2  or 
increase  both.  We  know  that  the  increase  in  conductivity 
comes  almost  entirely  from  increase  in  a,  the  degree  of  dis- 
sociation. For  we  saw  in  §181  and  Experiments  XVIII  and 
XIX  that  the  degree  of  dissociation  of  a  dissolved  substance 
increases  with  the  dilution,  and  we  know  that  the  fluidity 
(reciprocal  of  viscosity,  §127)  upon  which  U^  and  U2  must 
depend  does  not  change  appreciably.1  We  should  therefore 
expect  that  if  the  dilution  was  carried  far  enough,  a  point 
would  be  reached  where  all  the  electrolyte  would  be  disso- 
ciated (i.e.  a  =  i),  and  further  dilution  would  not  appreciably 
increase  the  conductivity.  Such  is  found  experimentally  to 

1  Spring.  Pogg  Ann.,  clix,  p.  i;   Wagner,  Zeit.  phys.  Chem.,  v,  p.  36. 


288   ELECTROCHEMISTRY,  ELECTROLYTIC  CONDUCTION. 


'OS^Of 

H 
H 

s 

0 

M 

cs 
n 

to 

rt- 

CO 

>OSuZf 

M 

1 

1 

1 

CN) 

(N 

•M^f 

H 

d 

M 

M 

00 

1 

00 

0 

>OS<* 

* 

IO 

* 

H 

to 

oq 

-COK>«« 

H 

d 

H 
M 

M 

s 

0 
csi 

WO<H-D 

00 

CO 

vd 

to 

d 

H 

rt- 

<0,0* 

M 
H 

00 

M 
H 

oq 

M 
H 

cs 

M 
H 

M 

So 

M 

eri 

I* 

M 
M 

d 

M 

oq 

s- 

H 

H 

M 

t/3 
D 

£ 

°> 

'ON*N 

vo 

O 
H 

5 

oq 

0 

M 

M 
ON 

rj- 

oo 

S 

1 

TJ 

s 

'ON* 

00 

10 

H 
H 

0 
to 

IH 
M 

M 
H 

oo' 
0 

M 

i 

oq 

0 

c 

J} 

m 

M 

d 

H 

2 

S" 

*? 

M 

00 
ON 

ctf 
_> 

w 

ITO 

1 

M 

oq 

0 

H 

oq 

0 

M 

ON 

2 

HO*HN 

00 

S5 

S 

•s 

vO 
ON 

S 

ON 
00 

to 
0 

HO*N 

H 

1 

8 

f 

O 

M 

0 

'OS^H? 

CO 
GO 

M 

I 

to 

M 

o 

•O'H'O 

S 

o" 

H 

M 

J 

4 

H 

° 

EONH 

0 

00 

JC 

g 

0 
to 

0 

M 

to 

NO 

I3H 

oo 

s 

0 

M 

to 

M 

0 

* 

UOI^-BJJ 
-U3DU03 

0 

8 

0 

H 

8 

M 

O 

H 

^ 

o 

X 


HI 


DEGREE    OF    DISSOCIATION. 


289 


be  the  case  with  a  large  number  of  electrolytes,  and  it  is 
undoubtedly  true  in  all  cases  where  the  nature  of  the  ions 
is  independent  of  the  concentration.  We  will  designate  by 
^  the  limiting  value  of  >1,  corresponding  to  infinite  dilution 
(a  =  i).  Hence,  by  Equation  1  86, 

^=  96330  (UT+Ua)  (186') 

Therefore,  dividing  Equation  186  by  186',  we  find  that  the 
degree  of  dissociation  when  the  equivalent  conductivity  is  >l,  is1 


Table  XXXVI,  compiled  from  the  observations  of  Loomis,2 
Kohlrausch  and  Maltby,3  and  Ostwald,4  illustrates  the  close 
agreement  between  the  values  for  the  dissociation  calculated 
from  the  lowering  of  the  freezing  point  and  those  calculated 
from  the  equivalent  conductivity. 

TABLE  XXXVI. 

Degree  of  Dissociation.     Potassium  Chloride. 


Concentration 

Fr.    Pt. 

Cond. 

.01 

.946 

•94 

.02 

•9i5 

.92 

•03 

•9 

.91 

•05 

.89 

.89 

.1 

.862 

.86 

•4 

.802 

.80 

Tartaric  Acid. 


Concentration 

Fr.    Pt. 

Cond. 

.0106 
.0200 

.26 
.18 

.27 
.20 

.0499 
.0997 

•13 
.09 

•»3 

.09 

1  Arrhenius   Zeit.  phys.  Chem.,  1887,  i,  631. 

2  Wied.  Ann.,  1894,  li,  p.  500. 

s  Wiss.  Abh.  Reichsanstalt,  iii,  151. 
4 Zeit.  phys.  Chem.,  1889,  iii,  p.  371. 


2QO       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

291.  Temperature  Coefficient. — If  the  temperature  of  an 
electrolyte  is  raised,  the  equivalent  conductivity  generally 
increases,  but  the  explanation  is  quite  different  from  that  for 
the  similar  effect  of  dilution.  The  velocity  of  the  ions  increases 
enormously  with  rise  of  temperature,  owing,  to  a  great  de- 
crease in  the  viscosity,1  while  the  dissociation  generally  de- 
creases.2 If,  as  a  first  approximation,  we  regard  the  increase 
of  conductivity  as  proportional  to  the  increase  of  temperature, 
we  may  write 

kt=kio[i+a(t-t0)]  (188) 

where  kto  is  the  specific  conductivity  at  a  standard  tempera- 
ture and  a  is  the  temperature  coefficient.  The  temperature 
coefficients  of  a  number  of  the  more  common  electrolytes 
are  given  in  Table  XXXVII.  The  conductivity  is  not,  how- 
ever, a  simple  linear  function  of  the  temperature  (see  §120, 
note,  and  §303),  and  Equation  188  and  these  temperature 
coefficients  give  only  the  approximate  resistance  at  different 
temperatures. 

TABLE   XXXVII. 
Temperature  Coefficient   (i8°).3 


HN03  

0163 

H2SO4  

0164 

HC1  

0165 

KOH  

0190 

KNOS  

O2I  I 

KI  

0212 

KBr  

O2l6 

KC10,   

02l6 

AgN03  
K€l  

02l6 
O2I7 

NH4C1  

O2  19 

K2SO4   

0223 

CuSO4  

022  =; 

NaCl  

0226 

Na2SO4    

0234 

ZnSO4  

025 

1  Thorp  and  Rodger,  Zeit.  phys.  Chem.,  1894.  xiv,  p.  361;  1896,  xix,  p.  323. 

2  A.  A.  Noyes  and  Coolidge,  Zeit,  phys.  Chem.,  1903,  xlvi,  p.  323;  Proc.  Am. 
Acad.,  1903,  xxxix,  p.  163. 

3  KohlrausJi  Sitz.  Ber.   Berlin  Akad.,   1901,  p.   1026;   1902,    p.  572;  Proc. 
RoyJ  Soc.,  1903,  !xxi,  p.  338.     Deguisne,  Strassburg  Dissertation,  1895, 


SPECIFIC   AND    EQUIVALENT    CONDUCTIVITIES. 


291 


PROBLEMS  XXII. 

1.  10    gr.  of  anhydrous    CuSO4  were  dissolved  and  diluted  until 
the  solution  at  18°  filled  a  tank  20  cm.  long  and  5  cm.  cross  section. 
The    resistance    between    the    ends    was    125    ohms.     Calculate    (a) 
specific  resistance,   (6)  specific  conductivity,  (c)  equiva-  c^ 
lent     concentration,    (d)    equivalent    conductivity,    (e) 
molecular  conductivity. 

2.  What  is  the  degree  of  dissociation  of  this  solution 
(Table  XXXV)? 

3.  If  the  above  tank  has  a  resistance  of  97  ohms  at 
28°,  what  is  the  temperature  coefficient  of  the  solution? 

4.  The   equivalent    conductivity   of   a   tenth-normal 
sodium  chloride  solution  is  92.5.      Calculate  (a)  specific 
conductivity,    (6)    specific  resistance,  (c)  the  resistance 
of  the  above  tank  when  filled  with  this  solution. 

EXPERIMENT  XLVI. 

Absolute  Determination  of  Specific  and  Equivalent 
Conductivities.     Temperature  Coefficient. 

The  conductivity  vessel  consists  of  a  glass  tube  about 
i  cm.  internal  diameter  and  i  5  cm.  long,  fitted  with 
rubber  stoppers,  through  which  pass  copper  rods.  A 
copper  disk  which  nearly  fills  the  interior  of  the  tube  is 
attached  to  one  end  of  each  rod  and  the  other  end  is 
provided  with  a  transverse  hole  for  a  wire  connector  and 
a  longitudinal  binding  screw  (see  Fig.  92),  or  a  spring 
connector  (§95)  is  attached. 

The  electrolyte  is  a  solution  of  copper  sulphate  of 
assigned  strength.  The  vessel  is  nearly  filled  with  the 
solution.  There  are  marks  on  the  edge  of  each  elec- 
trode and  three  similar  marks  on  the  tube.  The  elec- 
trodes are  first  placed  so  that  their  marks  exactly 
coincide  with  the  outside  marks  on  the  tube.  Both 
electrodes  must  be  covered  by  the  solution,  all  air 
bubbles  being  carefully  excluded.  The  vessel  is  then 
mounted  vertically  in  a  thermostat  (§81,  and  note, 
Experiment  XXIX)  and  when  the  temperature  is  con- 
stant, the  resistance  is  determined  as  described  in  §69. 
The  upper  electrode  is  then  lowered  until  its  mark 
coincides  with  the  middle  mark  on  the  tube,  and  the 
resistance  is  again  determined. 

The  difference  between  the  two  resistances  is  the 
resistance  of  a  column  of  the  solution  whose  length  is 
the  distance  between  the  two  upper  marks  on  the  tube, 
and  whose  cross  section  is  that  of  the  tube.  The  latter 
is  afterward  determined  by  very  carefully  filling  the 
tube  with  water  from  a  burette,  and  finding  the  exact 
volume  contained  between  two  of  the  marks.  The 


FIG.  92 


distance  between  the  marks  is  obtained  with  a  comparator  (§28)  or  a 
vernier  beam  caliper  (§26).  Several  measurements  should  be  made 
of  both  quantities.  The  conductivity  tube  should  now  be  surrounded 
by  ice  and  water,  and  when  the  temperature  has  become  constant, 
the  resistance  should  be  redetermined. 

Calculate  and  tabulate  for  both  temperatures  (i)  resistance,    (2) 


292    ELECTROCHEMISTRY,  ELECTROLYTIC  CONDUCTION. 

specific  resistance,  (3)  conductivity,  (4)  specific  conductivity,  (5) 
equivalent  conductivity,  (6)  molecular  conductivity;  and,  finally, 
calculate  (7)  the  temperature  coefficient  of  resistance  and  (8)  the 
temperature  coefficient  of  conductivity,  taking  o°  as  the  standard 
temperature. 

QUESTIONS. 

1.  Explain  use  of  (a)  alternating  current,   (b)  telephone,   (c)  two 
lengths  of  solution,    (d)   electrodes  of  the   same  metal  as  the   salt 
dissolved. 

2.  Calculate  the  resistance  of  a  tank  filled  with  the  solution  you 
have  used,  if  the  tank  and  electrodes  are  twenty  cm.  square  and 
the  electrodes  are  i  m.  apart  and  (a)  if  the  temperature  is  o°,  (6)  if 
the  temperature  is  50°. 

292.  Ratio  of  Ionic  Velocities.  —  The  sum  of  the  velocities 
of  the  positive  and  negative  ions  can  be  calculated  from  the 
equivalent  conductivity  and  the  degree  of  dissociation  (Eqs. 
186,  187).  We  shall  show  that  the  ratio  of  the  velocities  of 
the  two  kinds  of  ions  can  be  found  from  the  changes  in  concen- 
tration of  the  electrolyte  at  the  two  electrodes. 

Suppose  we  have,  first,  two  platinum  electrodes  dipping  in 
an  electrolyte  and  that  we  send  through  the  solution  96,530 
coulombs  of  electricity.  Consider  the  solution  in  the  vicinity 
of  the  anode.  One  gram  equivalent  of  anions  has  been  liber- 
ated. In  the  solution,  the  current  is  divided  between  the 
anions  and  cations,  each  kind  of  ion  carrying  a  portion  pro- 
portional to  its  velocity,  since  the  total  charges  on  the  two 
are  equal.  Therefore  the  mass  of  anions  carried  into  the 
vicinity  of  the  anode  is  the  fraction 


of  a  gram  equivalent,  u^  and  u2  being  the  velocities  of  the  ca- 
tions and  anions,  respectively.  The  net  loss  of  anions  is 
therefore 


The   mass   of   cations   which   have  left    the   vicinity  of  the 
anode  is 


RATIO    OF    IONIC    VELOCITIES.  293 

Therefore  there  is  a  loss  of  anions  and  cations  of 


of  a  gram  equivalent,  or  this  amount  of  electrolyte  disappears 
from  the  vicinity  of  the  anode  when  96,530  coulombs  traverse 
the  liquid.  Similar  reasoning  will  show  that  the  fraction 


of  a  gram  equivalent  of  electrolyte  disappears  from  the  vicinity 
of  the  cathode. 

Observe  that  the  ratio  of  these  two  fractions  is  the  ratio 
of  the  velocities  of  the  two  types  of  ions,  or,  the  losses  at 
the  two  electrodes  are  inversely  proportional  to  the  velocities 
of  their  respective  ions. 

u^  _    Loss  at  anode 
u2  ~  Loss  at  cathode 

This  law  was  discovered  by  Hittorf  ,*  who  named  the  above 
fractions  "transport  numbers"  since  they  represent  the  share  of 
transport  of  electricity.  The  transport  number  for  the  anion 
is  the  one  usually  given  in  tables,  and  is  generally  understood 
unless  the  transport  number  for  the  cation  is  specified.  We 
will  represent  the  (anion)  transport  number  by 


The  transport  number  for  the  cation  is  obviously 


i-N  .       . 

(190) 


u2         N 

From  Equations  186,  and  189  or  190  we  can  calculate  Ul  and 
U2-  Table  XXXVIII  gives  the  transport  numbers  of  the 
more  common  electrolytes  (see  also  §295  and  Table  XXXIX). 

Ann.   1853,   Ixxxix,  p.    177;   also,   "Fund.   Laws  of   Elec.   Cond.," 


Goodwin. 


294       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

TABLE  XXXVIII. 
Transport  Numbers  (20°). l 


gr.  mol. 

liter 

•5 

5- 

HC1                                  

.i6s 

•  I  7  3 

.176 

.2*8 

H2SO4             

.167 

.  IQO 

.2OO 

.270 

NaOH 

80 

82 

8^ 

KC1 

.  ^i 

CQ 

•  ^o 

•  ^o 

NaCl    .             

.60 

.62 

.6^ 

.64 

CaCl2  

•S6 

.60 

.67 

.68 

CuSO4 

62 

60 

7  2 

AgNO.,  

•53 

•51 

•49 

•47 

ZnSO4  

.64 

.78 

293.  If  the  anode  is  a  metal  of  which  the  electrolyte  is  a 
salt,  the  anions  are  not  liberated,  but  an  equivalent  of  metal 
cations  goes  into  s.olution.     Instead  of  a  loss  of  electrolyte  at 
the   anode   there  will  therefore  be  a  gain.     The  amount  of 
metal  dissolved  at  the  anode  will  equal  that  deposited  at  the 
cathode,  and  therefore,  before  substituting  in  Equation  189, 
we  must  find  what  the  loss  would  have  been  with  a  platinum 
electrode,  by  subtracting  from  the  apparent  gain  in  electrolyte 
at  the  anode  the  equivalent  of  the  cathode  deposit. 

294.  Direct  Determination  of  Ionic  Velocities. — The  velocities 
of  certain  ions  have  been  determined  directly  and  the  values 
obtained  agree  well  with  those  calculated  by  Equations  186 
and  189.     The  electrolyte  consisted  of  two  or  more  solutions 
in   series,    separated   by   either   a   jelly   or   gravity.     In   the 
methods  of  Whetham2  and  Masson  one  of  the  solutions  pos- 
sessed a  colored  ion,  and  the  progress  of  this  ion  was  watched 
and    its   velocity   determined.      Lodge3    followed    hydrogen 

1  Compiled  from  the  exhaustive  tables  in  Winkelmann,  1905,  IV,  i. 
2 Phil.  Trans.,  1893,  A-  clxxxiv,  p.  337;  1895,  ccxxxvi,  p.  507. 
3  Brit.  Ass.  Rep.,  1886,  p.  389. 


IONIC    CONDUCTIVITIES. 


295 


ions  by  their  decoloration  of  a  phenolphthalein  solution. 
Steele1  observed  the  change  in  refractive  index  as  the  ions 
migrated.  The  original  papers  should  be  consulted  for  details. 
The  actual  velocities  of  the  ions  are  of  interest,  but  are  not, 
however,  of  as  much  importance  as  the  ionic  conductivities  to 
which  they  are  proportional. 

295.  Ionic  Conductivities. — Returning  to  Equation  186', 
we  see  that  the  equivalent  conductivity  of  a  completely 
dissociated  solution  is  made  up  of  two  parts,  one  of  which, 
96,530  t/j,  is  contributed  by  the  positive  ions,  and  the  other 
portion,  96,530  U2,  is  contributed  by  the  negative  ions. 

TABLE  XXXIX. 
Ionic  Conductivities  (i8°).2 


Cations 

Anions 

H 

.    318 

OH  

K  . 
NH4  .  .  . 

iPb  .  .  . 
JBa  ... 
Ag  
JCd  ... 

•  •  64.7 
64- 
61.5 
55-9 
54- 
47-4 
47-2 

a  2°4  '. 

C4HS04  
C4HS06  v  .... 
NO,  

68.7 

66. 

63- 
62. 

38. 
39- 
59. 

T  7 

4.6  6 

C2HO4  

47- 

fef.::: 

46.1 

4.7      C 

C4H,06  
C7H,O,  .  . 

39- 
79. 

38. 

C!H!O! 

3  ?. 

These    two    components    of    the   equivalent  conductivity, 
Xx  ,  are  called  the  ionic  conductivities  jl  and  j2 


/.  ^=a(jl+j2)  (192) 

Kohlrausch  and  Ostwald3  have  demonstrated  that  the  ionic 

1  Phil.  Trans.,  1902,  A,  cxcviii,  p.  105. 

2  Compiled  from  the  excellent  tables  in  Van  Laar,  Chap.  Ill;  and  Winkel- 
mann  (1905),  IV,  i,  except  for  Cl  and  NO3  which  are  taken  from  Goodwin's 
paper  in  Phys.  Rev.,  1904,  xix,  p.  369. 

3  Gott.  Nach.,  1876,  p.  213;     Wied.  Ann.,  1898,  Ixviii,  p.  785;  also  in  "Fund. 
Laws  of  Elec.  Cond.,"  Goodwin. 


296       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

conductivity  of  a  particular  ion  is  independent  of  the  presence 
of  other  ions  if  the  solution  is  very  dilute,  and  that  this 
law  is  also  approximately  true  for  moderate  concentrations.1 
We  would  naturally  expect  the  ions  would  be  mutally  inde- 
pendent in  a  solution  which  was  so  dilute  that  the  electrolyte 
was  almost  completely  dissociated. 

The  ratio  of  the  ionic  conductivities  of  a  salt  is  the  ratio 
of  the  velocities  of  the  ions  at  great  dilution  and  from  this 
ratio  corresponding  transport  numbers  can  be  calculated  by 
Equation  190. 

EXPERIMENT  XL VII. 
Velocity  of  Ions. 

(A)  Determination  of  Sum  of  Velocities,  and  Degree  of  Dissociation3 

Fill  a  standardized  conductivity  vessel  (§70)  with  the  assigned, 
electrolyte  and  determine  the  resistance  (§69).  Dilute  the  electro- 
lyte until  the  concentration  is  about  .0005  and  redetermine  the 
resistance. 

Calculate  the  equivalent  conductivities  for  both  concentrations. 
The  dissociation  may  be  considered  complete  in  the  more  dilute 
solution.  The  ratio  of  the  two  equivalent  conductivities  will  there- 
fore equal  the  degree  of  dissociation  a  of  the  more  concentrated 
solution.  Finally,  calculate  (Ui  +  U2)  by  Equation  186. 

(B)  Determination  of  Ratio  of  Velocities.     Ionic  Conductivities. 

Place  in  a  beaker  two  porous  cups  of  about  75  c.c.  capacity  and 
pour  in  the  assigned  solution  until  the  level  is  the  same  in  the  two  cups 
and  outside  in  the  beaker.  In  each  porous  cup  place  a  plate  elec- 
trode of  the  metal  whose  salt  was  assigned.  The  electrode  which 
is  to  be  the  cathode  should  have  been  carefully  cleaned  before 
immersion  and  weighed  to  tenths  of  a  milligram  (§32). 

Connect  the  two  electrodes  to  storage-battery  terminals  through 
an  ammeter,  a  switch  and  a  variable  rheostat  (see  Fig.  93).  Be 
careful  to  connect  the  cathode  plate  to  the  negative  pole.  If  the 
electrolyte  is  a  copper  salt,  the  current  per  100  cm.2  of  cathode  should 
not  exceed  one  ampere,  and  with  a  silver  salt  the  current  density 
must  be  much  less.  When  the  current  has  continued  sufficient 
time  to  give  a  suitable  deposit  on  the  cathode  (for  example,  .05  gr. 
in  40  minutes),  remove  and  carefully  dry  the  cathode,  and  weigh 
it  to  tenths  of  a  milligram.  Measure  the  volumes  of  the  three 
solutions.  Take  equal  volumes  of  each  (e.g.,  25  c.c.)  and  determine 
the  amount  of  the  salt  in  each  solution  by  determining  the  amount 
of  metal.  If  copper  is  the  metal,  determine  the  amount  either  by 

iZeit.  phys.  Chem.,  1888,  ii,  p.  841. 

2  The  most  convenient  electrolyte  is  copper  sulphate  of  concentration  between 
.1  normal  and  normal.  Silver  nitrate  with  silver  electrodes  offers  no  difficulties 
if  the  current  density  is  low  and  the  time  correspondingly  greater,  but  the  porous 
cups  must  be  sufficiently  dense  to  prevent  appreciable  diffusion  during  the  in- 
creased time. 


VELOCITY    OF    IONS. 


297 


(a)  depositing  the  copper  on  a  platinum  cathode,  using  a  low- 
current  density  and  platinum  cathode  and  anode,  and  continuing 
the  current  until  all  the  copper  is  deposited,  or  (b)  determine  the 
copper  by  iodimetry.1  This  is  much  more  rapid  and  sufficiently 
accurate  for  a  laboratory  experiment.  .Acidify  each  solution  with 
acetic  acid,  add  an  excess  of  potassium  iodide,  and  titrate  with  .1  n 
sodium  thiosulphite.  Remembering  that  one  molecule  of  thio- 
sulphite  corresponds  to  one  molecule  of  copper,  calculate  the  con- 
centration of  each  solution  from  the  titrations  and  the  volumes. 


Rheostat 


FIG.  93. 

From  the  original  volumes  of  the  solution  in  the  porous  cups,  calcu- 
late the  total  amount  of  copper  in  each,  before  and  after  the  passage 
of  the  current.  Subtract  from  the  value  for  the  anode  cup  the 
amount  deposited  on  the  cathode  (§293). 

Finally,  from  the  ratio  of  the  losses  in  the  two  cups,  calculate  the 
ratio  of  the  velocities  (Equation  189)  and  the  transport  number 
(Equation  190). 

If  silver  is  the  metal,  the  amount  of  silver  in  each  solution  may 
be  obtained  by  electrical  deposition,  using  platinum  electrodes  and 
a  very  small  current  density,  or  by  volumetric  analysis  with  potas- 
sium or  ammonium  sulphocyanide.2 

Combining  the  results  of  (A)  and  (B)  calculate  the  ionic  con- 
ductivity and  the  actual  velocity  of  each  ion. 

QUESTIONS. 

1.  What  is  the  velocity  of  each  ion  in  cm.  per  hour? 

2.  What  would  be  the  average  velocity  of  each  of  the  ions  of  this 
solution  between    two    fine  wires   5  cm    apart,  if   the  difference  of 
potential  of  the  two  wires  was  500  volts? 

Button,  "Volumetric  Analysis,"  p.  176. 
2Treadwell,  Vol.  ii,  p.  540. 


298   ELECTROCHEMISTRY,  ELECTROLYTIC  CONDUCTION. 

3.  With  the  help  of  Table  XXXIX  and  Equation  186'  calculate 
the  velocities  in  cm.   per  sec.   of  hydrogen  and  hydroxyl  ions  for 
unit  potential  gradient. 

4.  Calculate  the  velocities  of    the  latter  ions  if    the  solution   of 
(2)  was  replaced  by  pure  water. 

296.  Relative  Velocities  of  Ions.— Table  XXXIX  shows 
that  hydrogen  is  the  swiftest  ion  and  the  hydroxyl  ion  comes 
next  in  order  of  speed.  Suppose  we  have  an  alkaline  solution, 
and  gradually  add  an  acid  solution,  for  example,  imagine 
HC1  solution  added  to  NH4OH  solution. 

Up  to  the  neutrallization  point  the  mixture  is  alkaline 
and  therefore  practically  none  of  the  hydrogen  added  remains 
in  ionic  form  (§258).  The  only  effect  therefore  is  the  substi- 
tution of  the  other  ion  of  the  acid,  Cl',  for  the  faster  moving 
OH'  ions  and  hence  the  conductivity  decreases  to  the  neutrali- 
zation point.  The  further  addition  of  acid  beyond  the 
neutralization  point  increases  the  .number  of  the  rapid  hydro- 
gen ions  and  thus  increases  the  conductivity.  Analogous 
effects  are  observed  when  a  solution  of  a  strong  base  is  added 
to  a  solution  of  a  strong  acid. 

PROBLEMS  XXIII. 

1.  Calculate  Van't  Hoff's  coefficient,  i,  for  in  solutions  of  AgNO3, 
ZnSO4,  CuSO4,  and    Na2SO4.      (Equations   112   and   187   and  Table 
XXXV.) 

2.  Hittorf  found  that   after  the   current   had  passed  through   a 
normal  solution  of  CuSO4,  ,29 58  gr.  of  copper  had  been  deposited 
on  the  cathode  while  the  solution  in  its  vicinity  had  lost  .2114  gr. 
of  copper.     Calculate  (a)  ratio  of  velocities;  (b)  transport  number. 
With    the    assistance    of    Table    XXXV    calculate     (c)    the    degree 
of  dissociation;  (d)  the  ionic  conductivities,  and  (e)  the  velocities  for 
unit  potential  gradient. 

3.  A    current    was   passed   through    a    solution    of    silver    nitrate 
between  silver  electrodes.      While  the  cathode  gained  96.6  mg.   of 
silver,  the  electrolyte  in  its  vicinity  lost  50  mg.  of  silver.     Calculate 
(a)  ratio  of  velocities  of  ions,  (6)  transport  number. 

4.  From  Tables  XXXV  and  XXXVIII  determine  (a)  the  degree 
of  dissociation  of  a  normal  solution  of    hydrochloric  acid ;   (b)   the 
ionic   conductivities;    (c)    the   actual    velocities   of    the    ions   under 
unit  potential  gradient. 

5.  Calculate  the  actual  velocity  of  the  acetic  acid  radical  in  cm. 
per  sec.,  for  one  volt  per  cm. 

6.  .in  HC1  is  gradually  added  to  .in  NaOH,  until  the  volume  of 
the  solution  is  three  times  the  initial  volume.     Construct  a  curve 
which  will   illustrate   qualitatively   the  change  in  conductivity  (two 
values  may  be  obtained  from  Table  XXXV). 


ELECTRICAL  MEASUREMENT  OF  RATE  OF  SAPONIFICATION.      299 

7.  From  Tables  XXXV  and  XXXVII  calculate  the  temperature 
at  which  a  .01  normal  solution  of  CuSO4  has  the  same  specific  con- 
ductivity as  a  .001  normal  solution  of  Pb  (NO3)2  at  18°? 

EXPERIMENT  XLVIII. 
Conductivity  of  Acid,  Alkaline,  and  Neutral  Solutions.1 

As  an  illustration  of  the  difference  in  the  velocity  of  various  ions, 
we  shall  follow  the  change  in  resistance,  or  conductivity,  of  an  alkaline 
solution  while  an  acid  is  gradually  added. 

Fill  a  conductivity  vessel  (§70)  with  the  assigned2  alkaline 
electrolyte  so  that  the  electrodes  are  well  covered.  Add  a  few  drops 
of  phenolphthalein.  Determine  the  resistance  as  described  in 
§69,  preferably  using  a  slide-wire  bridge.  Add  from  a  burette  a 
small  measured  amount  of  the  assigned  acid  solution  and  redeter- 
mine  the  resistance.  Add  more  acid  and  again  determine  the 
resistance,  and  so  continue  until  well  beyond  the  neutralization  point. 
There  should  be  at  least  five  observations  on  each  side  of  this  point. 

In  the  vicinity  of  the  neutralization  point,  add  the  acid  slowly  so 
that  a  determination  of  the  resistance  may  be  made  as  close  as 
possible  to  this  point.  Plot  the  observed  resistances  against  the 
total  volume  of  acid  solution  added,  and  draw  a  smooth  curve 
through  the  points.  The  curve  should  show  a  sharp  change  of 
curvature  at  the  neutralization  point. 

This  method  of  determining  the  neutralization  point  is  sometimes 
useful  in  determining  the  strength  of  an  acid  or  alkaline  solution 
which  is  so  turbid  or  colored  that  an  indicator  cannot  be  observed. 

QUESTIONS. 

1.  From  considerations  of  the  progressive  changes  in  ionic  con- 
centrations   deduce    the  form  of  curve  obtained  when   (a)   a  weak 
acid  is  added  to  a  weak  base;    (6)  when  a  strong  acid  is  added  to  a 
weak  base ;      (c)  when  a  strong  base  is  added  to  a  weak  acid.      Explain 
why  the  change  of  curvature  is  indistinct  in  these  three  cases. 

2.  From  the  constant  of  the  cell  (§72)   calculate  (a)  the  specific 
conductivity  of  the  neutral  salt  at  the    neutralization    point,   and 
(6)  its  equivalent  conductivity. 

297.     Electrical  Measurement  of  Rate  of  Saponification.— 

In  §244  we  saw  that  when  an  alkali  is  added  to  a  solution  of 
an  ester  both  gradually  disappear  and  the  corresponding  ace- 
tate and  alcohol  are  formed.  In  experiment  XXXIII  we 
followed  the  reaction  by  determining,  at  intervals,  from  titra- 
tions,  the  unchanged  alkali. 

From  the  previous  paragraphs  it  is  evident  that  the  dis- 
appearance of  the  rapidly  moving  hydroxyl  ions  of  the  base, 

1  Findlay,  "Practical  Physical  Chemistry,"  p.  179. 

2  A  solution   of   any  one  of   the  more  common  strong  bases  is  satisfactory 
and  the  particular  acid  is  immaterial.     It  is  convenient  to  have  the  strength  of 
the  acid  solution  about  twice  that  of  the  alkaline  solution. 


300   ELECTROCHEMISTRY,  ELECTROLYTIC  CONDUCTION. 

and  the  substitution  of  the  little  dissociated  alcohol  and  acetate 
for  the  highly  dissociated  base,  must  decrease  the  conductivity, 
and  Walker  has  shown  *  that  the  conductivity  is  very  closely 
proportional  to  the  amount  of  base  unchanged. 

EXPERIMENT  XLIX. 
Study  of  Saponification  by  Change  in  Conductivity. 

Prepare  a  conductivity  vessel  and  Wheatstone's  bridge  so  that  a 
resistance  determination  may  be  made  immediately  after  filling  the 
conductivity  vessel.  The  latter  should,  preferably,  be  in  a  thermo- 
stat (§81).  Prepare  the  assigned2  solutions  of  ester  and  base  and, 
noting  the  exact  time,  mix  the  two  solutions.  Immediately  fill 
the  conductivity  vessel  with  the  mixture  and  determine  the  resist- 
ance (§69). 

About  three  minutes  later  make  another  observation  of  the 
resistance.  Notice  the  exact  time  when  a  balance  is  secured.  So 
continue,  gradually  increasing  the  intervals.  The  last  observation 
but  one  should  be  about  three  hours  after  the  first,  and  the  final 
determination  after  about  twenty-four  hours. 

Plot  observed  resistances  against  the  time  and  draw  a  smooth 
curve  through  the  observations.  Choose  five  points  on  the  curve 
and  calculate  the  velocity  constant  for  each. 

If  the  initial  concentrations  were  equal,  Equation  1  59  must  be 
used  and  x/c  —  x  for  any  point  is  the  difference  between  the  ordinate 
at  that  point  and  the  initial  ordinate  divided  by  the  difference  be- 
tween this  ordinate  and  the  final  ordinate. 

If  the  initial  concentrations  are  unequal,  Equation  161  must  be 
used.  c2,  the  smaller  concentration,  is  proportional  to  the  total 
change  in  resistance.  At  any  one  of  the  points  in  the  curve  (c2  —  x) 
is  proportional  to  the  change  in  resistance  beyond  that  point.  Ci 
is  proportional  to  the  resistance  corresponding  to  c2,  multiplied 
by  the  ratio  of  the  initial  concentrations,  and  x  for  any  point  is 
proportional  to  the  change  of  resistance  at  that  point. 

Thus  all  the  factors 

c2,  (GI  -x),  clt  (c2  -x) 

may  be  expressed  in  terms  of  resistances.  The  actual  concentrations 
must  of  course  be  substituted  in  the  factor  (d  —  c2). 

Find  the  mean  velocity  constant  and  compare  it  with  the  value 
obtained  in  Experiment  (XXXIII),  if  identical  solutions  have  been 
employed. 


.  Royal  Soc.,  1906-7,  Ixxviii,  p.  157. 
2  Ethyl    acetate    is  the  most  convenient  ester,   and  the  solution  may  have 
any  strength  between  .O2«  and  .iw.     The  alkali  may  be  any  one  of  the  more 
common  bases  and  its  concentration  may  be  the  same  as  that  of  the  ester  or 
slightly  different. 


EQUIVALENT  CONDUCTIVITY  FOR  INFINITE  DILUTION.       301 

QUESTIONS. 

1.  (Time  constant.)  Determine  from  the  curve  when  the  reaction 
was  half  completed,  and  compare  it  with  the  time  calculated  from 
the  velocity  constant  (Equation  155). 

2.  What  was  the  initial  velocity  of  the  reaction?     In  what  units 
is  your  answer  expressed? 

3.  What   advantages  and  what   disadvantages  has  this  method 
over  that  of  Experiment  XXXIII? 


Determination    of    the    Equivalent    Conductivity  for   Infinite 

Dilution. 

298.  The  most  obvious  method  of  determining  the  equiva- 
lent conductivity  at  infinite  dilution  is  to  determine  it  for  a 
large  number  of  concentrations,  the  last  being  as  dilute  as 
possible,  and  then,  by  exterpolating,  determine  what  would  be 
the  conductivity  at  infinite  dilution.  A  convenient  method  of 
exterpolating  is  to  plot  the  observed  conductivities  against 
the  concentrations  (or,  preferably,  the  cube  root  of  the  con- 
centration, since  this  will  give  a.  more  convenient  scale)  and 
extend  the  curve  obtained  until  it  cuts  the  axis  of  conductiv- 
ities at  zero  concentration.  This  method  is  applicable  to 
acids,  bases,  and  salts  which  are  strongly  ionized.  , 

Some  electrolytes  are  so  little  dissociated  that  it  is  impossible 
to  determine  the  equivalent  conductivity  for  infinite  dilution 
in  this  manner.  In  such  a  case  we  must  find  some  highly 
dissociated  combination  of  each  one  of  the  ions,  and  determine 
the  ionic  conductivities.  By  Kohlrausch's  law  (§295)  the 
equivalent  conductivity  for  complete  dissociation  is  always 
equal  to  the  sum  of  the  ionic  conductivities  of  the  constituent 
ions.  For  example,  acetic  acid  is  only  dissociated  to  a  small 
degree  at  great  dissociation.  We  can,  however,  find  the  ionic 
conductivity  of  the  hydrogen  ion  from  the  equivalent  conduc- 
tivity and  transport  number  of  HC1  at  great  dilution,  and  we 
can  similarly  find  the  ionic  conductivity  of  the  other  ion,  the 
acetic  acid  radical,  by  measurements  upon  sodium  acetate 
which  is  highly  dissociated. 


302   ELECTROCHEMISTRY,  ELECTROLYTIC  CONDUCTION. 

The  Dilution  Laws. 
299.  We  saw  in  §260  that 

K= — 

V(i  -  a) 

where  V  is  the  volume  (litres)  which  contains  one  gram 
equivalent,  i.e.,  the  dilution,  a  is  the  degree  of  dissociation, 
and  K  is  a  constant  called  the  dissociation  constant  (Ostwald's 
dilution  law,  §260).  Substituting  for  a  its  value  given  by 
Equation  187,  we  have 

K=~      ;—  (193) 


If  a  is  very  small,  we  may  neglect  it  in  comparison  with 
unity  and 

:.K  =  ~  (194) 

Equation  193  holds  well  for  weak  electrolytes,  but  when  we 
apply  it  to  strong  electrolytes  we  find  that  K  is  not  a  constant 
but  depends  upon  the  dilution.  It  is  almost  impossible 
to  believe  that  the  law  of  mass  action,  upon  which  this  equation 
is  based,  is  inapplicable  to  strong  electrolytes.  It  seems 
probable  that  there  are  forces  between  the  ions  themselves,  and 
between  the  ions  and  the  neutral  molecules  of  strong  electro- 
lytes, which  make  it  impossible  to  apply  the  mass  law  directly 
to  the  ions  and  molecules.  The  papers  of  Jahn1  and  Nernst2 
should  be  consulted  for  a  full  discussion  of  the  subject. 

300.  Two  empirical  modifications  of  the  above  dilution  law 
apply  quite  satisfactorily  to  strong  electrolytes.  In  place  of  the 
first  power  of  the  dilution,  V,  Rudolphi's^  formula  substitutes 
the  square  root  of  the  dilutiojn  and  Kohlrausch's4  formula  has 

1  Zeit.  phys.  Chem.,  1901,  xxxvii,  p.  490;  1902,  xli,  p.  257. 
2Zeit.  phys.  Chem.,  1901,  xxxviii,  p.  487. 
s  Zeit.  phys.  Chem.,  1895,  xv»>  P-  385. 
4  Zeit.  phys.  Chem.,  1895,  xv»i>  662. 


DISSOCIATION    CONSTANT.  303 

the  cube  root.  Noyes  and  Coolidge1  found  that  the  latter 
formula 

K  =  v*T~(T  (ig^ 

I/MX  (Xoo  —  A) 

gave  values  of  K  which  were  constant,  within  the  experimental 
errors,  for  the  highly  dissociated  salts  KC1  and  NaCl,  through- 
out a  wide  range  of  concentrations  (.3  to  100  normal),  and 
an  extreme  range  of  temperature  (i  8°  to  306°)  . 

301.  Dissociation  Constant.  —  The  dissociation  of  an  electro- 
lyte is  often  quite  complex  and  may  take  place  in  stages. 
For  example,  sulphuric  acid  may  dissociate  as  follows:2 

H2SO4^±H'  +  HSO4';   HSO/  =  H-  +  SO/ 
and  water  may  dissociate  as 


In  such  cases  there  will  evidently  be  two  dissociation  constants 
which  will  not  be  independent,  since  they  involve  common 
ions. 

As  has  been  previously  pointed  out,  all  the  different  methods 
of  measuring  the  strengths  of  acids  or  bases  give  values 
which  are  proportional  to  the  dissociation.  The  dissociation 
constant  is  perhaps  the  most  convenient  measure  of  the  dis- 
sociation, and  is  therefore  of  great  importance.  It  is  also 
often  called  the  ionization  constant,  and,  for  reasons  just 
explained,  another  name  for  it  is  affinity  constant. 

TABLE  XL.3 

Dissociation  Constants  (25°) 

Acetic  acid  .........................  oooo  1  8 

Benzoic  acid.  .  :  .....................  000073  •* 

Succinic  acid  .......................  000066 

Formic  acid  ........................  0002  14 

Tartaric  acid  .......................  00097 

Salicylic  acid  .......................  00102 

Ammonia  ..............  -.  ...........  000023 

*lx. 

2Perkin,  "Electrochemist,"  1901,  i,  p.  189. 
3Landolt  and  Bernstein's  Tables. 
4Bauer,  Zeit.  ph.  Chem.,  1906,  Ivi,  p.  215. 


304       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

302.  Isohydric  Solutions. — Two  solutions  are  called  iso- 
hydric  if  the  total  number  of  each  kind  of  ion  is  unchanged 
upon  mixing.  We  will  only  consider  one  case,  namely,  the 
mixture  of  two  binary  electrolytes  which  have  a  common  ion ; 
for  example,  two  univalent  acids.  Let  the  dilution  of  one 
solution  be  V  (volume  occupied  by  one  gram  equivalent)  and 
its  degree  of  dissociation  a  (=  fraction  of  gram  equivalent 
of  each  ion  in  volume  V)  and  let  the  corresponding  quantities 
for  the  other  solution  be  V  and  a'.  If  the  electrolytes  are 
weak,  we  may  apply  Oswald's  dilution  law  and  for  the 
first  electrolyte  we  must  have 

K-          2 


V(i-a) 

After  mixing,  the  total  volume,  or  dilution,  is  V -\-V'y  and  if 
the  ions  are  to  be  unchanged,  the  amount  of  the  common  ion  in 
this  volume  is  (a  +  a') . 

The  dissociation  constant  of  the  first  solution  must  therefore 
also  agree  with  the  equation 

(a+a'}a 


Dividing  one  equation  by  the  other,  we  have 

(a+a')V  =          .     a+a'  ^V  +  V 
(V+W)a~  a  V 

a       a' 


(196) 
V       V 


or  two  binary  electrolytes  having  a  common  ion  are  isohydric, 
if  they  have  equal  concentrations  of  the  common  ion.1 

1  For  an  excellent  discussion  of  isohydric  solutions,  see  Walker,  4th  Ed.  Chap. 
XXVI. 


THE    DILUTION    LAW."  305 

PROBLEMS  XXIV. 

1.  Velocity  df  reaction   between  n/2$    solutions  of   ethyl   acetate 
and  sodium  hydroxide.     Change  in  electrical  conductivity. 

Time  (min.)  .Resistance 

o 270 

3 3°° 

M 35° 

22 390 

44 425 

75 452 

240 520 

720 567 

Calculate  (a)  velocity  constant,  (6)  initial  velocity  of  the  reaction, 
(c)  time  when  the  reaction  is  half  completed. 

2.  Calculate  from  Table  XXXIX  the  equivalent  conductivity  at 
infinite  dilution  of   (a)  oxalic  acid;    (6)  tartaric  acid;  (c)  ammonium 
hydroxide. 

3.  Calculate  the  degree  of  dissociation  of  a  .005  normal  solution 
of  acetic  acid  from  the  dissociation  constant  (Table  XL). 

4.  Calculate  (a)  the  degree  of  dissociation  of  a  .001  normal  solution 
of  tartaric  acid  (Eq.  176  and  Table  XL);  (6)  the  equivalent  conduc- 
tivity; (c)  the  specific  resistance,  and  (d)  the  resistance  of  a  tube  of 
this  solution  i  metre  long  and  i  square  decimeter  cross  section. 

5.  Calculate  both  Rudolphi's  constant  and  Kohlrausch's  constant 
for  a  low  solution  of  (a)  HC1;  (6)  NaOH. 

6.  We  have  a  .1  normal  solution  of  potassium  chlorate.      Estimate 
from   Table    XXXV   and    Equation    196   the   concentrations  of  iso- 
hydric  solutions  of  (a)  KI;  (6)  K2SO4. 

7.  From   Tables   XXXIX   and   XL   calculate    (a)    the   degree   of 
dissociation;    (b)    the   molecular   conductivity,    and    (c)    the   specific 
resistance  of  a  i  %  solution  of  benzoic  acid. 


EXPERIMENT  L. 

The  Dilution  Law. 

Prepare  with  great  care  a  sufficient  quantity  of  a  .05  normal 
solution  of  the  assigned  electrolyte.1  Choose  such  a  pipette  that 
the  standardized  conductivity  vessel  (§§70,  72),  may  be  properly 
filled  with  two  transfers  of  the  pipette.  Determine  the  conductivity 
of  the  solution,  following  the  directions  given  in  §69. 

By  means  of  this  pipette  remove  half  of  the  solution  and  in  its 
place  introduce  from  a  similar  pipette  an  equal  amount  of  pure 
water.  Determine  the  resistance  as  before.  Continue  replacing 
half  of  the  solution  by  water  and  determining  the  resistance  until 
the  concentration  is  1/256  of  what  it  was  initially. 

For  each  observation,  calculate  (i)  the  concentration  in  gram 
equivalents  per  litre;  (2)  the  specific  conductivity;  (3)  the  equivalent 
conductivity;  (4)  the  degree  of  dissociation  (Equations  187  and  191) 
and,  finally,  (5)  the  dissociation  constant. 

i  Suitable  electrolytes  are  any  of  those  whose  constants  are  given  in  Table  XL. 

20 


306       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

QUESTIONS. 

1.  (a)  Calculate  the  number  of  grams  of  (a)  undissociated  solute 
(6)  of  each  ion,  in  one  litre  of  the  original  solution. 

2.  For  each  ion  calculate  (a)  the  ionic  conductivity,  and  (b)  the 
actual  velocity  for  unit  potential  gradient. 

3.  Could  one  determine  ^  by  exterpolation  ?     Explain. 

303.  Electrolytic   Conductivity   and   Temperature. — It   was 

pointed  out  that  Equation  188  is  only  approximate.  Kohl- 
rausch1  found  that  he  could  express  the  equivalent  conductivity 
for  infinite  dilution,  at  any  temperature  t,  in  terms  of  the 
conductivity  at  18°,  by  the  empirical  equations 

l*^ ^ I8[i+a(t-i&)+b(t -i&)*];  b  =  .oi63(a-. 0174),  (197) 
Upon  substituting  the  values  of  ^  l8,  a,  and  b,  which  he  had 
determined  for  a  great  number  of  electrolytes,  he  found  that 
^<x>*  would  be  equal  to  zero  at  some  temperature  between  —37° 
and  —41°,  for  every  electrolyte.tested. 

It  is  interesting  to  note  that  the  empirical  formula  for  the 
viscosity  of  pure  water  at  different  temperatures  (§127)  gives 
a  minimum  within  this  range  of  temperature. 

304.  Hydrates. — This   coincidence   suggests  that   each   ion 
in  an  aqueous  solution  attracts  a  number  of  water  molecules, 
which  more  or  less  completely  surround  the  ion  and  that  the 
resistance  to  the  motion  of  the  ion  is  therefore  simply  the 
viscous  resistance  between  bodies  of  pure  water.     Further 
evidence  of  the  formation  of  such  hydrates  is  given  in  the 
papers  of  (Nernst2  Buchbock,^  Jones,4  and  Washburn.5     The 
latter  has  shown  that  the  minimum  hydration  ranges  from  .3  of 
a  molecule  of  water  for  hydrogen  to  4.7  molecules  for  lithium. 

305.  Conductivity  at  High  Temperatures  and  Pressures.— 
Noyes  and  his  co-workers6  have  measured  the  conductivity  of 
aqueous  solutions  of  the  common  salts,  acids,  and  bases  over  a 
range  of  temperature  extending  to  about  300°.     The  solutions 
were  kept  liquid  by  great  pressure.     A  platinum-lined,  steel 

1  Berichte,  Ber.  Ak.,  1901,  p.  1026;  1902,  p.  572;  Proc.  Roy.  Soc.,  1903,  338. 

2G61t,  Nach.,  1900,  pp.  i  and  68. 

sZeit.  phys.  Chem.,  1906,  Iv,  p.  563. 

4  Am.  Chem.  Journal,  1905,  xxxiv,  p.  294;  1907.  xxxvii,  p.  126;  1909,  xli,  p.  19. 

s  Jour.  Am.  Chem.  Soc.,  1909,  p.  322. 

6Z.  c. 


CONDUCTIVITY    OF    NON-AQUEOUS    SOLUTIONS.  307 

bomb  was  used  and  the  joints  were  made  tight  with  gold 
bushings.  Quartz  was  used  where  an  insulating  material  was 
required.  They  found  that  the  conductivity  was  very  great 
at  high  temperatures.  The  conductivity  for  infinite  dilution 
(estimated  by  Kohlrausch's  Equation  195)  showed  even  a 
greater  increase,  and,  therefore,  by  Equation  187,  the  dissocia- 
tion must  have  decreased.  Moreover,  the  limiting  value  of 
/lx,  as  the  temperature  was  increased,  was  approximately 
the  same  for  all  binary  electrolytes  (about  noo).  The 
conductivity  generally  increases  as  the  temperature  rises, 
because  the  velocity  of  the  ions  increases  more  rapidly  than  the 
dissociation  decreases.  (Compare  Equation  185.) 

This  explanation  suggests  the  possibility  of  some  tempera- 
ture where  the  dissociation  begins  to  decrease  more  rapidly 
than  the  velocity  of  the  ions  is  increasing.  This  would  result 
in  a  maximum  of  conductivity  at  this  temperature.  Such 
is  undoubtedly  always  the  case  near  the  critical  temperature. 
Noyes  and  Coolidge1  found  such  maxima-  for  fairly  concen- 
trated aqueous  solutions  of  sodium  and  potassium  chloride. 
Kraus2  has  demonstrated  such  a  temperature  maximum 
for  ethyl  and  methyl  alcohol  solutions,  and  Hagenbacks  for 
solutions  in  sulphur  dioxide.  A  few  solutions  show  such  a 
maximum  of  conductivity  at  moderately  low  temperatures, 
e.g.,  copper  sulphate  at  95°.4 

306.  Conductivity  of  Non-aqueous  Solutions. — A  great 
amount  of  work  has  been  done  upon  the  electrical  conductivity 
of  solutions  in  other  solvents  than  water,  but  it  is  only  possible 
to  give  a  brief  summary.  For  more  detailed  information, 
the  reader  should  refer  to  the  original  memoirs  of  Walden,5 
Kraus6  and  Kahlenberg.7 

'Proc.  Am.  Acad.,  1903,  xxxix,  p.  163. 

2  Phys.  Rev.,  1904,  xviii,  pp.  40,  89. 

3  Ann.  de  Phys.,  1901,  v,  p.  276. 

4  Sack,  Ann.  d.  Phys.,  1891,  xliii,  p.  212. 

s  Zeit.  f.  Anorg.  Chem.,  1900,  xxv,  p.  209;  1902,  xxix,  p.  371;  xxx,  p.  149.  Zeit. 
f.  phys.  Chem.,  1903,  xliii,  pp.  396,  464;  xlvi,  p.  103;  1906,  liv,  p.  129;  Iv,  pp. 
207,  281,  683. 

6  Am.  Chem.  Jo.,  xxiii,  p.  277;   Jour.  Am.  Chem.  Soc.,  xxvi,  p.  499;  xxvii,  p. 
101. 

7  Jour.  Phys.  Chem.,  1902,  vi,  p.  447;  1903,  vn,  p.  254. 


308       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 


Liquid  sulphur  dioxide,  liquid  ammonia,  and  hydrocyanic 
acid  are  the  best  known  inorganic  solvents,  while  among 
organic  solvents,  ethyl  and  methyl  alcohol,  pyridene,  and 
formic  acid  have  been  carefully  studied. 

Solutions  in  all  these  solvents  show,  in  general,  the  electrical 
characteristics  of  aqueous  electrolytes.  The  equivalent 
conductivity  of  a  dissolved  substance  increases  with  the 
dilution,  indicating  an  increase  in  the  dissociation. 

TABLE   XL!.' 

Degree  of  Dissociation,  a,  of  i/io  Normal  Solutions  of  Potassium 
Iodide  and  Potassium  Acetate  in  Water,  and  Ethyl,  and  Methyl  Alcohol ; 
and  Equivalent  Conductivity  at  Infinite  Dilution,  ^  . 


K 

[ 

KC2 

H302 

*QC 

a 

^ 

« 

H2O 

I2O   7 

88 

OO  0 

8^ 

CH3OH  

95-2 

•52 

77.8 

•36 

C2H5OH 

48  o 

2  ^ 

12   8 

16 

The  dissociation  even  at  great  dilution  is  however,  generally 
much  less  than  in  aqueous  solutions,  but  the  equivalent  con- 
ductivity is  often  much  greater.  We  see  by  Equation  186  that 
the  velocity  of  one  or  both  the  ions  must  be  much  greater  than 
in  solutions  in  water.  Kraus  has  shown  that  in  liquid  am- 
monia solutions  the  anion  is  apparently  an  electron  (§345), 
which,  on  account  of  its  diminutiveness,  travels  with  great 
speed.  Solutions  in  liquid  ammonia  obey  the  dilution  law 
(§299).  Non-aqueous  solutions  often  do  not  show  the  same 
degree  of  dissociation  by  the  freezing-point  method  (Eq.  112) 
and  by  the  conductivity  (Eq.  187);  and  Kohlrausch's  law 
(§295)  usually  fails.  They  resemble  aqueous  solutions  (§305) 
in  showing  an  increase  of  conductivity  as  the  temperature 
is  raised  until  a  certain  temperature,  beyond  which  the  con- 
ductivity falls  off. 

1  Vollmer,5Wied.  Ann.,  1894,  Hi,  p.  328;  and  Jones,  Zeit.  phys.  Chein.,  1899, 
xxxi,  p.  114. 


CONDUCTIVITY    OF    FUSED    SALTS. 


309 


Skaupy  has  applied  Oswald's  dilution  law  to  solutions  of 
metals  in  mercury  (amalgams)  and  has  shown  that  the  ions 
are  similar  to  the  ions  which  Kraus  found  in  ammonia  solutions, 
namely,  the  anion  is  an  electron  and  the  cation  is  the  metal.1 

For  the  conductivity  of  salts  in  mixed  solvents,  the  reader 
is  referred  to  the  papers  of  Jones  and  his  co-workers.2 

307.  The  Conductivity  of  Fused  Salts. — Faraday  discovered 
that  salts  which  conduct  electricity  when  dissolved  in  water 
also  conduct  when  they  are  melted,  even  though  no  water  is 
present,  and  that  they  obey  his  law  (§28i).3 

The  specific  conductivity  (§10)  of  a  fused  salt  may  greatly 
exceed  the  specific  conductivity  of  the  most  concentrated 
solutions,  but  the  concentration  is  so  high  that  the  equivalent 
conductivity  is  much  less. 

Goodwin  and  Mailey4  studied  not  only  the  conductivity, 
but  also  the  fluidity  (§127)  of  a  large  number  of  the  more 
common  salts  in  the  molten  state,  and  they  found  that 
conductivity  and  fluidity  increased  as  the  temperature  rose, 
but  that  the  fluidity  increased  somewhat  faster,  showing  that 
the  increase  of  conductivity  was  due  to  an  increase  in  the  ve- 
locity of  the  ions  rather  than  to  an  increase  in  the  dissociation.5 

TABLE  XLII.6 
Fused  Silver  Nitrate. 


Temp. 

k 

A 

F 

218  (Melt.pt.) 

.681 

29.2 

250 

.834 

36.1 

27.7 

300 

1.049 

46.2 

36-7 

350 

1.245 

55-4 

45-5 

'Zeit.  phys.  Chem.,  1908,  Iviii,  p.  580. 

2  Am.  Chem.  Jo.,  1902,  xxviii,  p.  329;  1904,  xxxii,  pp.  409,  521;  1906,  xxxvi, 
pp.  325,  427. 

3Exp.  Res.,  iv,  380;  vii,  669. 

4 Phys.  Rev.,  1908,  xxvi,  p.  28. 

s  See  also  the  papers  of  Arndt;  Zeit.  f.  Electrochemie,  xii,  p.  337;  xiii,  p.  59; 
xiv,  p.  662. 

6  Goodwin  and  Mailey,  /.  c. 


3IO       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

Table  XLII  gives  the  specific  conductivity,  k,  the  equivalent 
conductivity,  X,  and  the  fluidity,  F,  of  fused  silver  nitrate  at 
different  temperatures.  The  specific  conductivity  of  a  60% 
aqueous  solution  at  18°  is  but  .208. 

308.  Electrolytic  Conductivity  of  Solids. — Hot  non-metallic 
solids,    such    as    glass,   porcelain,    etc.,  conduct  appreciably. 
Warburg1    inclosed   a   glass   plate   at   300°   between   sodium 
amalgam  electrodes  and  found  that  it  conducted  electricity, 
and  that  the  products  liberated   (Na2SiO3  at  anode)  were  in 
accordance  with  Faraday's  law.     If  a  lithium  amalgam  was 
used  the  progress  of  the  colored  lithium  ions  through  the  glass, 
could  be  observed.     Practically  all  the  current  was  carried  by 
the  cations,  sodium,  or  lithium.     Warburg  found  that  a  hot 
crystal  of  quartz   showed   enormously  greater  conductivity 
parallel  to  the  axis  than  at  right  angles.     Nernst  considers  that 
the  conductivity  of  the  metallic  oxide  glower,  which  he  invented 
and   which   is   used   in    Nernst   lamps,    is   electrolytic.     He 
explains  the  small  amount  of  decomposition,  -even  with  direct 
current,  by  the  great  rapidity  of  diffusion  of  the  products  of 
the  electrolysis  at  the  high  temperature  employed.2 

309.  Dielectric    Constants. — The    attraction    between    the 
two  ions  of  a  binary  molecule  is  given  by  the  fundamental 
electrostatic  equation 

F-e- 

~td* 

Where  F  is  the  force  in  dynes,  e  is  the  ionic  charge  in  electro- 
static units,  d  is  the  distance  between  the  ions  in  centimeters, 
and  e  is  the  dielectric  constant  (specific  inductive  capacity) 
of  the  medium  surrounding  the  two  ions.  The  greater  e,  the 
less  the  attractive  force,  and  therefore  we  should  expect  that 
the  higher  the  dielectric  constant  of  a  solvent,  the  easier  the 
ions  of  a  molecule  would  separate,  and  the  less  frequently 
would  they  recombine;  in  other  words,  the  greater  would  be 
the  dissociation. 

'Ann.,  1884,  ii,  p.  622. 

2Zeit.  f.  elec.  Chem.,  1899,  vi,  p.  41. 


DIELECTRIC    CONSTANTS. 


TABLE  XLIII.i 
Dielectric    Constants. 


IJ 


Hydrocyanic  acid 96 

Water 80 

Methyl   alcohol 33 

Ethyl    alcohol 25 

Ammonia  (liquid) 22 

Acetone 17 

Sulphur    dioxide 14 

Pyridene 12 


Ether 4.5 

Xylol 2.26 

Benzol 2.2 

Toluol 2.2 

Petroleum 2.07 


310.  Nernst -Thomson  Rule. — In  general,  such  is  the  case, 
as  is  illustrated  in  the  following  table.  The  liquids  in  column 
I  give  conducting  solutions  while  those  in  column  II  do  not. 
All  the  values  are  either  for  static  charges  or  for  very  low 
frequencies.  While  the  dielectric  constant  is  evidently  a 
very  important  factor  in  determining  the  dissociation,  there 
are  other  modifying  influences  which  are  occasionally  more 
important.  For  example,  the  strong  acids  conduct  better  in 
hydrocyanic  acid  than  in  water,  while  the  reverse  is  true  of 
potassium  salts.2 

PROBLEMS  XXV. 

1.  Calculate  the  equivalent  conductivity  of  a  .001  normal  solution 
of  ZnSO4  at  o°.      Use  Kohlrausch's  formula  (197)  and  the  value  of  a 
given  in  Table  XXXVII. 

2.  How   would   you   interpret    Noyes'    observation   that    at    high 
temperatures   and   great    dilution   the   equivalent    conductivities   of 
all  binary  electrolytes  approach  a  common  value  ? 

3.  What  is  the  transport  number  for  (a)  the  anion;  (b)  the  cation 
in  the  ammonia  solution  of  Kraus  described  in  §306? 

4.  Compare  the   degree   of  dissociation  given  in   Table    XXXVI 
with  the  dielectric  constants  as  given  in  Table  XLIII. 

5.  Calculate   (a)  the  concentration;   (6)  the  specific  resistance  of 
melted  silver  nitrate  at  300°  (Table  XLII,  Density  of  fused  salt  =4.3  5). 

6.  How  many  milligrams  of  sodium  silicate  should  Warburg  have 
found  (§308)  after  the  passage  of  i  milliampere  for  an  hour? 

7.  Two  parallel  plates,  12  cm.  in  diameter  are  separated  by  2  mm. 
of  benzol.     Calculate  the  force  drawing  them  together  when  their 
difference  of  potential  is  (a)    10  electrostatic  units;   (6)    1000    volts 

1  From  Winkelmann,  iv,  i,  p.  135. 

2  Kahlenberg,  Jour.  Phys.  Chem.,  1902,  vi,  p.  447. 


312       ELECTROCHEMISTRY,    ELECTROLYTIC    CONDUCTION. 

(§  §10,  12  and  309)  calculate  the  force  on  unit  charge  from  the  poten- 
tial (§§9,  10),  and  the  charge  from  the  capacity  (§9). 

EXPERIMENT  LI. 

Determination  of  the  Dielectric  Constant  of   a  Liquid   by  DeSauty's 

Method. 

Determine  the  dielectric  constant  of  a  highly  insulating  liquid 
such  as  kerosene  or  benzol  by  the  method  described  in  §80. 

Make  repeated  careful  determinations  of  the  proper  resistances 
for  the  largest  distance  between  the  plates  which  will  give  a  distinct 
minimum  of  the  sound.  Then  surround  and  cover  the  plates  with 
the  liquid  and  again  determine  the  proper  resistances.  Check  your 
observations  by  changing  the  resistances  back  to  their  values  for  air 
and  then  adjusting  the  distance  between  the  plates  for  minimum 
sound.  The  ratio  of  the  distances  apart  of  the  plates  should  be 
(approximately)  inversely  proportional  to  the  ratio  of  the  resistances 
when  the  distance  was  kept  constant. 

QUESTIONS. 

1.  Calculate  the  force  in  dynes,  between  two  ions  .01  mm.   apart 
in  this  liquid,  if  the  charge  on  each  is  3  X    io-10  electrostatic  units. 

2.  Calculate  the  capacity  of  the  two  plates  when  separated  by 
(a)  air;  (6)  liquid,  the  distance  apart  being  the  same  as  in  the  first 
part  of  the  experiment  (§9). 

3.  Calculate  the  charge  for  each  case  if  (a)  100  electrostatic  units  of 
potential  are  applied;  (6)  100  volts  (§§9,  12). 


CHAPTER  IX. 


ELECTROCHEMISTRY.     POTENTIAL  DIFFERENCES. 

311.  Difference  of  Potential  Between  a  Metal  and  a  Solution 
of  One  of  its  Salts. — A  metal  dips  in  a  solution  of  one  of  its 
salts.  The  ions  in  the  solution  tend  to  increase  their  volume; 
i.e.,  they  exert  a  pressure  which  we  call  the  osmotic  pressure,  p 
(§169.)  Nernst  has  suggested1  that  we  may  consider  that 
there  are  also  ions  in  the  metal  and 
that  they  are  under  a  similar  pressure 
which  he  calls  the  solution  pressure,  and 
which  we  will  designate  by  P.  If  P  is 
greater  than  p,  ions  will  go  from  the 
metal  into  the  solution,  and  since  metal 
ions  have  a  positive  charge  (§285-287), 
the  solution  will  become  positively 
charged  (Fig.  94).  Since  the  removal 
of  a  positive  charge  is  equivalent  to 
the  addition  of  a  negative  charge,  the 
metal  will  become  charged  negatively. 
This  negative  charge  will  attract  the 


FIG.  94. 


po'sitive  metallic  ions  which  will  be  repelled  by  the  positively 
charged  solution.  Equilibrium  will  be  attained  when  this 
electric  force  opposing  the  solution  of  the  metal,  is  equal  to 
the  force  equivalent  to  P  —p,  which  tends  to  produce  solution. 
We  will  show  that 


(198) 


where  E  is  the  potential  of  the  solution  above  the  metal,  in 
volts,  J  is  the  number  of  joules  in  one  calorie   (4.187,    §7), 
v  is  the  valency  of  the  metal,  m  is  the  number  of  ions  in  a 
'Zeit.  phys.  Chem.,  1889,  iv,  p.  129. 


314        ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

molecule,  Q  is  the  heat  absorbed  from  surrounding  bodies 
during  the  transfer  of  this  electricity,  k  is  a  factor  defined 
later,  R  is  the  gas  constant  (in  calories),  and  0  is  the  absolute 
temperature  .  P  and  p  have  already  been  defined  . 

312.  At  equilibrium,  the  energy  of  the  metal  ions  must  be  the 
same  in  metal  and  solution  (equal  thermodynamic  potentials,  §235) 
and  a  transfer  of  ions  between  the  two  requires  no  work. 

The  energy  gained  by  the  system  when  one  gram  molecule  of  ions 
is  carried  from  the  metal  to  the  solution  is  the  charge,  96,5301^, 
multiplied  by  E,  or 

joules  (§10) 


The  mechanical  energy  lost  by  the  system,  when  one  gram  molecule 
of  ions  is  transported  from  the  pressure  P  to  the  lower  pressure  p  is 


Cpdv  =  -    Cvdp  =  -kRO  C^P-  =  kROln  —  (198') 

«y  •/  *  »/        P  i 


if  we  assume  that  Boyle's,  Gay  Lussac's,  and  Van't  Hoff's  laws  are 
applicable.     For,  since  the  temperature  is  constant, 

pv=  const.     .'.  pdv=—vdp 

According  to  Van't  Hoff's  law  (Equation  in),  the  equation  of  state 
for  one  gram  molecule  of  solute  is 


Since,  however,  we  are  considering  the  ions  in  both  solvent  and 
electrode,  we  shall  use  the  constant  k  which  we  will  define  later. 
Furthermore,  to  make  the  case  general  we  will  let  Q  be  a  quantity 
of  heat  which  is  absorbed  from  neighboring  bodies  during  this  isother- 
mal transfer  of  one  gram  molecule.  If  we  express  the  three  forms 
of  energy  in  joules,  the  electrical  energy  gained  must  equal  the 
mechanical  energy  lost  plus  the  heat  energy  absorbed,  or, 

p 
kROln +7^-96530  vmE=o 

.'.£— ^-       -(/£  +  kROln - 

P  \ 

(198) 


For  convenience,  we  have  considered  one  gram  molecule  of  ions. 
So  great  a  transfer  would  usually  disturb  the  equilibrium  and 
properly  we  should  have  considered  a  small  fraction,  w,  of  a  gram 
molecule.  This  factor  would,  however,  cancel  from  the  equations. 


NULL    ELECTRODES.  315 

313.  In  order  to  test  this  equation,   we  must  find  some 
electrode  which  will  acquire  the  potential  of  the  electrolyte. 
If  metal  and  electrolyte  are  initially  uncharged,  and  p=P, 
there  will  be  equilibrium  between  the  ions  in  each  when  they 
come  together,  and  the  difference  of  potential  will  be  zero.     P 
must  therefore  equal  the  osmotic  pressure,  p,  of  a  solution  of 
such  concentration  that  no  charges  and  no  difference  of  poten- 
tial appear  when  the  two  come  in  contact. 

314.  Null  Electrodes. — Metallic  ions  of  the  more  "noble" 
metals  are  usually  deposited  on  the  metal  from  the  solution; 
for  example,  if  a  copper  plate  is  placed  in  a  solution  of  copper 
sulphate,  copper  ions  will  leave  the  solution  even  if  it  is  very 
dilute,  and,  depositing  on  the  copper,  give  it  a  positive  charge. 
But  if  we  add  potassium  cyanide,  the  almost  undissociated 
salt  K2Cu2(CN)4  is  formed  and  by  the  addition  of  a  proper 
amount,  the  concentration  of  copper  ions  may  be  reduced  until 
the  metal  and  solution  are  at  the  same  potential.     Further 
addition  of  KCN  will  cause  solution  of  the  copper,  giving  it  a 
negative  charge. 

By  this  means,  the  concentration  of  the  copper  ions  may 
be  so  reduced  that  the  copper  is  even  more  negative  to  the 
solution  than  zinc  is  to  a  solution  of  one  of  its  common  salts. 
Mercury  is  another  noble  metal  whose  cyanide  salt  solution 
may  have  an  osmotic  pressure  equal  to  its  own  solution 
pressure,  so  that  there  is  no  difference  of  potential  between 
metal  and  solution.  Such  a  combination  is  called  by  Billitzer 
a  null  electrode.1 

315.  Dropping   Electrodes. — A   second   means   of   reducing 
the  concentration  of  the  ions  of  a  noble  metal,  until  the  metal 
and  the  solution  are  at  the  same  potential,  is  to  exhaust  the 
ions  by  presenting  fresh  increasing  surfaces  as,  for  example, 
in  the  dropping  electrode  of  Kelvin  and  Hemholtz.2 

The  dropping  electrode  consists  of  a  very  fine  capillary 
through  which  a  stream  of  the  metal  flows  into  the  electrolyte. 
The  extremely  fine  stream  breaks  into  individual  drops  at 

JAnn.  der  Phys.,  1903,  ii,  p.  902;  Freundlich  and  Makelt,  Zeit.  elect.  Chem., 
1909,  xv,  p.  161. 

2Paschcn,  Ann.  der  Phys.,  1890,  xliv,  p.  42. 


316         ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 


the  surface  of  the  electrolyte.  The  mercury  ions  in  the  solution 
are  deposited  upon  the  fresh  expanding  surfaces  of  the  drops, 
giving  the  drops  a  positive  charge.  This  positive  charge 
attracts  the  negative  ions  of  the  salt 
(electric  double  layer}  and  drags  them 
down  with  itself.  When  a  drop 
reaches  the  mercury  in  the  bottom, 
its  surface  and  capacity  are  reduced 
and  some  of  the  mercury  ions  leave  the 
mercury  and  unite  with  the  negative 
ions  which  were  carried  down. 

This  exhaustion  of  the  salt  at  the 
end  of  the  mercury  column  continues 
until  the  osmotic  pressure  of  the  re- 
maining ions  is  equal  to  the  solution 
pressure  of  the  metal,  and  the  mer- 
cury, both  in  the  stream  and  in  the 
reservoir,  has  the  potential  of  the 
solution.  If,  now,  we  measure  the 
difference  of  potential  between  the 
mercury  in  the  reservoir  (A)  (see 
Fig.  95)  and  the  mercury  in  the 
IFlG  95  bottom  of  the  electrolyte  (F),  we  shall 

obtain    the     difference    of    potential 
between  mercury  and  this  particular  solution. 


EXPERIMENT  LIL 
Dropping  Electrodes. 

A  dropping  electrode  will  be  used  to  determine  the  difference 
of  potential  between  mercury  and  a  saturated  solution  of  calomel. 

A  funnel  is  drawn  out  to  the  finest  point  which  will  permit  a  flow 
of  mercury  when  the  funnel  is  nearly  full.  A  platinum  wire  is  sealed 
into  the  bottom  of  a  large  test-tube  or  tall  beaker  so  that  only  a 
very  short  length  projects  inside.  Sufficient  mercury  is  poured 
into  the  bottom  to  cover  the  platinum,  and  the  vessel  is  nearly 
filled  with  a  normal  solution  of  potassium  chloride  to  which  an 
excess  of  calomel  has  been  added.  The  funnel  is  mounted  so  that 
the  capillary  tip  just  touches  the  solution  and  is  filled  with  mercury, 
connection  to  which  is  made  by  a  platinum  wire.  The  two  platinum 
wires  are  connected  to  a  compensation  apparatus  (§73),  and  their 
difference  of  potential  is  determined,  when  the  stream  has  run  for 


ELECTROCAPILLARY    PHENOMENA. 


317 


a  sufficient  time  for  a  steady  state  to  be  reached.  The  result  obtained 
will  probably  be  low  on  account  of  the  errors  due  to  the  limitations 
of  facilities  and  time  which  are  imposed  upon  a  laboratory  exercise. 

316.  Electrocapillary     Phenomena. — Another    method     of 
reducing  the  mercury  ions,  so  that  the  metal  and  electrolyte 
are  at  the  same  potential,  is  to  force  the  ions  from  the  solution 
upon  the  mercury  by  the  electric  current.     The  electrolyte 
is  in  a  fine  capillary  tube  so  that  new  ions  only  slowly  diffuse 
in  from  the  rest  of  the  solution. 

Under  ordinary  circumstances,  mercury 
ions  are  deposited  upon  mercury  and  give  it 
a  positive  charge,  and  the  electrolyte  a  nega- 
tive charge.  Since  the  like  charges  repel  and 
the  opposite  charges  attract,  both  tend  to 
increase  the  surface;  that  is,  they  oppose  the 
surface  tension.  Therefore,  when  the  surface 
tension  effects  are  a  maximum,  the  mercury 
and  solution  must  be  at  the  same  potential. 

Since  glass  is  not  wet  by  mercury  (Table 
XV)  the  surface  tension  will  cause  it  to  con- 
tract in  a  capillary,  and  the  maximum  con- 
traction will  be  obtained  when  the  surface 
tension  is  unopposed  by  the  electric  forces. 

317.  Let  us  see  how  the  position  of  the  mercury  meniscus  in 
the  capillary  will  change  as  we  vary  the  applied  electromotive 
force.     Let  E=  difference  of  potential  between  mercury  and 
electrolyte,    C=  capacity   of   surface   per   square   centimeter. 
The  electrical  energy  of  unit  area  is  one-half  the  product  of 
the  potential,  E,  and  the  charge,  CE,  or  i/2CE*.     The  surface 
tension   is  numerically   equal   to   the   energy   per  unit   area 
(§122).     Let  r  =  surface  energy  when  the  difference  of  poten- 
tial is  E  and  Tm  =  maximum  or  true  surface  energy  or  tension. 

.'.  T=Tm-\CE*  (199) 

If  T,  which  is  proportional  to  the  contraction  in  the  capillary, 
is  plotted  against  E,  the  curve  will  evidently  be  a  parabola, 
whose  maximum  corresponds  to  the  point  E=O.  This  assumes 


FIG.  96. 


318        ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

that  Tm  is  independent  of  E,  which  is  not  strictly  true,  but  the 
correction  is  small  and  will  be  found  in  specific  treatises.1 

318.  Standard  Calomel  Electrode. — If  by  either  of  these 
last  two  methods,  the  dropping  or  the  capillary  electrode, 
we  make  the  difference  of  potential  between  the  electrode 
and  the  electrolyte  zero,  and  if  we  measure  the  potential 
difference  between  this  electrode  and  any  other  in  contact 
with  the  electrolyte,  the  observed  difference  of  potential  will 
be  that  between  this  second  metal  electrode  and  the  electrolyte. 
Having  in  this  manner  obtained  the  potential  difference  between 
some  particular  electrode  and  its  solution,  we  may  use  this 
electrode  as  a  standard.  A  very  convenient  standard  is  the 
calomel  electrode  which  is  described  in  §75.  Its  difference  of 
potential  is  usually  taken  as 

E=  .s6o[i  +  .0006  (/  —  18°)] 
for  a  normal  solution,  and  for  a  decinormal  solution 

E  =  .6i3[i  +  .oooo8(/-  1 8°]) 
The  mercury  is  of  course  positive. 

TABLE  XLIV.' 
Potentials  of  Electrodes  in  Normal  Solutions  of  their  Ions  (18°). 

(The  positive  sign  signifies  that  the  electrode  is  at  a  higher  poten- 
tial than  the  electrolyte.) 


Electrode 

Polential  (volts) 

Electrode 

Polential  (volts) 

K  . 

—  2  t; 

Hg2  . 

+  1.027 

-Na  

'•  j 

2  . 

HK.. 

......      +1.048 

Zn 

0 

+  i  306 

Cd 

I  A  3 

Cl 

+  i  604. 

Ni  . 

OAQ 

NO, 

+  1.7=; 

Pb  
H  

•w^y 
+   .129 

+      277 

OH  
SO4  

+i-94 

+2.2 

Cu 

+      606 

HSO4 

+  2    Q 

'Van  Laar,  Chap.  XII;  Whetham,  Chap.  XI. 

2  Largely  from  Wilsmore  and  Ostwald,  Zeit.  phys.  Chem.,  1901,  xxxvi,  p.  92; 


CALOMEL    ELECTRODE.  319 

These  values  are  probably  too  high,1  and  therefore  the 
potential  differences  given  in  Table  XLIV  which  are  based  on 
these  values  are  probably  somewhat  in  error,  but  the  relative 
values,  which  are  far  more  important  than  the  absolute  values, 

quite  accurate. 

In  Table  XLIV  the  ions  of  the  electrode  are  present  in 
ie  electrolyte  in  normal  concentration,  except  in  the  case 
>f  O  and  OH,  where  the  value  given  is  for  normal  concentra- 
:ion  of  hydrogen  ions.  Many  of  the  values  have,  however, 
>een  exterpolated  by  Equation  202  from  observations  at  lower 
mcentrations. 

319.  The  gas  and  radical  electrodes  are  either  platinum-black 
(§71)   electrodes  saturated  with   the  gas   (for  example,    the 

tydrogen  electrodes  of  Experiment  LVI)  or  compounds  from 
which  these  ions  are  liberated.  For  example,  the  calomel 
electrode,  Hg  +  Hg2Cl2,  reversibly  receives  or  gives  up  chlorine 
ions.  When  chlorine  ions  arrive,  mercury  dissolves.  When 
they  leave,  mercury  is  precipitated.  Such  a  reservoir  of 
negative  ions  is  called  an  electrode  of  the  second  class,  while 
positive  electrodes  are  said  to  be  of  the  first  class. 

320.  The    capillary    electrometer2    which    depends    upon 
the  displacement  of  mercury  in  a  capillary  is  a  very  convenient 
and  sensitive  instrument  for  detecting  small  potential,  differ- 
ences.    It  is  described  in  §74. 

PROBLEMS  XXVI. 

1.  Why   must   the   mercury    stream   from   a    dropping    electrode 
break  into  drops  inside  the  electrolyte? 

2.  Would    it    be    possible    for   the    electrolyte    about    a  dropping 
electrode   to   become    so    exhausted    that   the    mercury   acquired   a 
potential  below  that  of  the  solution? 

3.  Calculate    the    difference    of    potential    between    (a)    a    "null 
electrode."  and  a  zinc  rod  dipping  in  a  normal  zinc  solution;  (6)  an 
electrode  of  mercury  which  drops  into  a  normal  silver  solution  and 
a  silver  rod  dipping  in  the  same  solution ;  (c)  a  lead  plate  in  a  normal 
lead   solution   and   a   column   of   mercury   showing   maximum   con- 
traction in  a  capillary  tube  filled  with  the  solution. 

(The  potential  difference  between  solutions  may  be  neglected 
in  this  and  the  following  problems.) 

rSmith  and  Moss,  Phil.  Mag.,  1908,  xv,  478;  Palmaer,  Zeit.  phys.  Chem.,  1907, 
iv,  p.  129. 
2Lippmann;  Ann.  der  Phys.,  1873,  cxlix,  p.  547. 


320        ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

4.  An  earthed  copper  rod  dips  in  a  normal  copper  solution.      If 
the  solution  is  joined  by  a  syphon  successively  to  normal  solutions 
of  silver,  lead,  mercury,  and  zinc,  find  the  potentials  of  rods  of  (a) 
silver;  (b)  lead;  (c)  mercury,  and  (d)  zinc,  dipping  in  the  respective 
solutions. 

5.  A  normal  calomel  electrode  dips  in  a  normal  zinc  solution  and 
a  zinc  rod  dips  in  the  latter.      What  is  the  difference  of  potential 
between  the   zinc  and  the  mercury?      (b)    What   would   have  been 
the  difference  of  potential  if  the  rod  and  the  solution  had  been  of 
copper? 

EXPERIMENT  LIII. 
Potential  Difference  Between  Mercury  and  Sulphuric  Acid. 

Capillary  Electrometer. — Set  up  a  capillary  electrometer  (§74) 
with  a  scrupulously  clean  new  capillary  tube  and  fresh  pure  (1:6) 
sulphuric  acid  solution  and  mercury.  Connect  it  to  a  compensation 
apparatus  (§73)  through  a  key  with  upper  and  lower  contacts  which 
keeps  the  electrometer  short-circuited  except  when  depressed  (see 
Fig.  2oa  and  §74).  Make  the  mercury  in  the  capillary  negative  and 
apply  various  electromotive  forces. 

For  each  potential,  observe  the  position  of  the  meniscus  and  also 
its  position  before  and  after  applying  the  e.m.f.  Subtract  the  mean 
of  the  two  latter  from,  the  reading  when  the  potential  is  applied. 
Gradually  increase  the  applied  e.m.f.  until  the  deflection  (above 
difference)  begins  to  decrease.  Also,  make  several  observations 
of  the  position  of  the  meniscus  for  very  low  voltages,  when  the 
mercury  in  the  capillary  is  positive. 

Plot  your  results,  making  applied  e.m.f.'s  abscissae  and  deflections 
ordinates.  Determine  from  your  curve  the  natural  potential 
difference  between  mercury  and  the  sulphuric  acid  solution. 

QUESTIONS. 

1.  Why   must   potentials   which   give   continuous   electrolysis   be 
absolutely  excluded   (i.e.,  potentials  several  tenths  of  a  volt  above 
that    corresponding   to    maximum    deflection,    or   which    make    the 
mercury  positive  by  several  tenths  of  a  volt)  ? 

2.  Is  the  curve  obtained  a  parabola?     Explain. 

ENERGETICS  OF  A  CELL. 

321.  Heat  of  Reaction. — Let  us  further  consider  Equation 
198.  We  have  seen  that 

kRdln— 
P 
is  the  mechanical  work  accompanying  the   solution   of  one 

gram  molecule  of  the  electrode.  If  no  electrical  energy  is 
produced,  this  work  which  accompanies  the  solution  appears 
as  an  equivalent  amount  of  heat  and  in  Chapter  IV,  we  have 
seen  how  this  can  be  determined. 


CONCENTRATION    CELLS.  321 

There  remains  the  term  Q  which  represents  the  heat  exchange 
with  the  surroundings.  We  shall  first  consider  several  cases 
where  this  term  is  eliminated. 

322.  Concentration  Cells.  —  If  we  have  two  soluble  electrodes 
of  the  same  material,  dipping  in  either  the  same  or  different 
connected    electrolytes,  the    passage  of  the  current  will  be 
accompanied    by   solution    of   one    electrode    and    an    equal 
deposition  upon  the  other,  and,  therefore,  if  we  find  the  dif- 
ference of  potential  of  the  two  electrodes,  Q  will  be  plus  for 
one  and  negative  for  the  other  and  will  cancel  out. 

323.  (A)   The  two  electrodes  are  of  different  concentrations 
but  dip  in  a  common  electrolyte.     For  example,  the  two  elec- 
trodes may  be  amalgams  of  different  concentrations  or  plati- 
num-black saturated  with  gas  under  different  pressures.     The 
resultant   electromotive  force  may  be   found  by  considering 
Equation  198  for  both  electrodes.     We  shall  use  the  prefix  i 
for  one  electrode  and  2  for  the  other,  remembering  that  the 
two  differences  of  potential  oppose  each  other  and  that  the 
solution  is  the  same  for  both,  and  hence  p  is  common 

E  =          =  1.99x4.187x2.30  M  i  ,    P,  _  j    PA 

96530?^  p  P  ' 


k  c 

=  1.98X10-4  —  01og-  (200) 

v^n  c 


For  the  solution  pressures  in  the  two  electrodes  will 
evidently  be  proportional  to  the  concentrations  ct  and  c2. 
k  was  the  provisional  factor  in  the  equation  for  one  gram 
molecule  of  ions  (§312).  In  this  case  we  are  only  concerned 
with  the  ionic  state  in  the  electrodes.  Let  us  assume  that 
k  =  i  .  Such  an  assumption  would  seem  natural  for  electrodes 
consisting  of  absorbed  gas  (since  it  is  unity  in  the  gaseous 
state,  Equation  57).  The  following  representative  example 
shows  that  it  is  also  permissible  with  amalgams. 


322         ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

The  difference  of  potential  between  two  mercury-zinc 
amalgams  of  concentrations  .003366  and  .00011305  was 
found  to  be  .0419  volts  at  u.60.1  From  equation  200, 

E=  i.gSX  lo-4 —  log  —  =  i.pSX  10-*—    -  log  29.7  =  .0416 

WYl  C  2  2 

(Vapor    pressure    measurements    have    shown    that    zinc    is 
monatomic  (m  =  i)  when  dissolved  in  mercury.) 

324.  (B)  Two  identical  electrodes  in  solutions  of  the  same 
electrolyte  of  different  concentrations.  Let  £x  =  potential 
of  the  electrode  in  the  more  concentrated  solution,  E2=  po- 
tential of  the  other  electrode  above  its  solution,  and  £3=the 
difference  of  potential  between  the  two  solutions.  Since 
the  electrodes  are  identical,  P  is  the  same  in  each  and  cancels 
out.  Moreover,  the  osmostic  pressures  pt  and  p,  are  pro- 
portional to  the  concentrations  ct  and  c2  (§§169-171).  Van't 
Hoff's  equation  for  solutions  (§173)  tells  us  that  if  we  consider 
one  gram  molecule  of  solute  k=i  and  m  =  2,  for  both  the  posi- 
tive and  the  negative  charges  are  included  in  the  gram  mole- 
cule of  solute. 


E  =  E,  -  E2  -  £3  =  i  .98  X  10-4—0  log  —  -  E3     (200') 

2  "V  C i 


Determination  of  the  Difference  of  Potential  Between  Two  Similar 
Electrolytes  of  Concentrations  d  and  C2. 

325.  Let  u-i  be  the  velocity  of  the  cation  and  u2  that  of  the  anion. 
When  one  gram  molecule  of  ions  is  dissolved  at  one  electrode  and 
deposited  at  the  other,  the  share  transported  between  the  two 
solutions  by  the  cation  is  ullul  +  u2,  and  that  by  the  anions,  in  the 
other  direction,  is  u2/ul  +  u2  (§292). 

If  the  electrodes  are  of  the  same  substance  as  the  principal  cation 
of  the  electrolyte,  the  current  due  to  the  difference  of  concentration 
will  flow  from  the  weak  to  the  strong  solution,  in  the  electrolyte, 
for  it  must  equalize  the  concentration  [principle  of  stable  equilibrium 
(§188)].  The  excess  transported  by  the  cations  is  fti— ffe/ttr+tta  of  a 
gram  molecule,  and  since  the  cations  travel  with  the  current,  it  is 
carried  from  the  weak  concentration,  Ci,  to  the  strong  concentration, 
c2  (see  Fig.  97). 

1  G.  Meyer,  Ann.,  1892,  xvi,  p.  292. 


CONCENTRATION    CELLS. 


323 


The   work  represented   by  this  transfer   is    (Equation    198'  and 
§§323»  324)- 


2.30  log    — 


But 


Ul  **2  XT    XT-.  -y 

=  i  —  2N  (Eq.  1 90) 

.'.  E3=  i.gSX  10-4(1  —  2Af) — log  —  (201) 


©- 


0 


,C2 
© 


FIG    97. 


Observing  that  E3  opposes  the  potential  difference  due  to 
the  electrodes,  Equation  200'  becomes 


Ni0 1       c2 
E  =  i  .98  X  10-4 log  — 


(202) 


If  the  electrodes  are  of  the  second  class  (§319),  E3  must  be 
added  instead  of  subtracted. 

325a.  If  the  potential  difference  between  a  metal  and  a 
solution  is  known  for  one  concentration  c^  (e.g.,  Table  XLIV), 
the  potential  difference  for  any  other  concentration  c2  may  be 
found  by  substituting  in  Equation  202  and  adding  (alge- 
braically) the  result  to  the  potential  for  the  concentration  cl. 


324        ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

EXPERIMENT  LIV. 

Difference  of  Potential  of  Metal  and  Electrolyte. 
Influence  of  Concentration. 

Clean  carefully  two  electrodes  of  the  assigned  metal *  and  prepare 
the  two  assigned  solutions.  Place  an  electrode  in  a  beaker  or  tumbler 
and  pour  in  one  of  the  solutions.  Connect  a  normal  calomel  elec- 
trode (§75)  to  the  electrolyte  (if  necessary,  using  an  intermediate 
electrolyte)  and  determine  the  difference  of  potential  between  the 
metal  and  the  mercury  in  the  calomel  electrode  by  the  potentiometer 
method  (§73  and  Fig.  21).  After  all  the  adjustments  are  completed, 
and  before  the  final  determination,  it  is  well  to  clean  the  electrode 
and  stir  the  solution.  Observe  carefully  which  electrode  is  positive 
and  follow  the  directions  regarding  the  addition  or  subtraction  of 
the  potential  of  the  calomel  electrode  (§76).  Find  similarly  the 
potential  difference  for  the  other  concentration.  Finally,  place 
one  electrode  in  a  porous  cup  and  place  the  cup  in  a  tumbler  or 
beaker.  Put  the  other  electrode  in  the  tumbler,  outside  the  cup. 
Pour  one  solution  into  the  porous  cup  and  the  other  into  the  tumbler 
outside.  Determine  the  difference  of  potential  of  the  two  electrodes 
by  the  potentiometer  method.  . 

In  the  report,  give  the  difference  of  potential  between  the  metal 
and  each  concentration,  the  difference  in  these  values,  the  difference 
as  determined  directly  and  the  difference  as  calculated  by  Equation 
202.  i  should  be  calculated  from  Equations  112  and  187  together 
with  the  data  given  in  Table  XXXV.  An  approximate  value  of  N 
may  be  obtained  from  Table  XXXVIII. 

QUESTIONS. 

1.  What    would    have    been    the    difference    of    potential    of    the 
electrodes  at  (a)  o°;  (b)  ioo°? 

2.  Why  does  this  difference  of  potential  change  upon  standing? 

3.  What  would  be  the  change  in  the  concentrations  of  the  two 
solutions  if  the  cell  delivered   one-hundredth  of  an  ampere  for  one 
hour? 

326.  Determination  of  Solubility. — We  can  determine  the 
concentration  of  a  very  slightly  soluble  salt  by  measuring 
the  electromotive  force  of  a  concentration  cell,  one  of  the 
solutions  of  which  contains  this  salt.  For  example,  suppose 
we  wish  to  determine  the  solubility  of  silver  chloride  at  a 
given  temperature.  We  set  up  a  cell  with  two  silver  electrodes 
separated  by  a  porous  cup.  In  the  cup  we  pour  a  saturated 
solution  of  silver  chloride,  outside  the  cup  a  known  solution 
of  some  soluble  salt,  for  example  .oin  AgNO3.  The  observed 
electromotive  force  is  substituted  in  Equation  202  and  the 

JSilver,  copper,  zinc,  and  lead  are  satisfactory  metals  and  the  electrolyte 
may  be  .in  and  .oiw  concentrations  of  any  of  their  common  salts. 


DETERMINATION    OF    SOLUBILITY.  325 

unknown  concentration  CL  is  calculated.  If  the  porous  cup 
also  contains  a  known  solution  of  a  soluble  chloride  salt, 
for  example,  .in  KC1,  the  concentration  of  the  chlorine  ions, 
c',  is  approximately  known.  If  C  is  the  concentration  of  the 
dissolved  silver  chloride. 


KC  is  called  the  solubility  product. 

If  we  dissolve  silver  chloride  in  pure  water  the  concentra- 
tion of  both  ions  will  be  the  same  and  will  equal 

\/KC  =\fcj' 

This  will  also  equal  approximately  the  concentration  of  salt 
dissolved  or  the  solubility,  for  such  a  very  insoluble  salt  is 
almost  completely  dissociated. 

EXPERIMENT  LV. 
Determination  of  the  Solubility  of  a  Salt  of  Low  Solubility. 

If  silver  chloride  is  the  salt,  a  few  drops  of  silver  nitrate  added  to 
a  .in  potassium  chloride  solution  will  give  a  solution  saturated  with 
silver  chloride,  without  seriously  disturbing  the  chlorine  concen- 
tration. If  iodide  of  potassium  is  substituted  for  the  chloride,  the 
concentration  of  silver  iodide  may  be  determined. 

The  solubility  of  lead  sulphate  may  be  determined  by  setting  up 
a  cell  with  two  lead  electrodes,  one  of  which  is  in  a  porous  cup.  Out- 
side the  cup  is  a  solution  of  .oin  Pb(NO3)2,  and  in  the  cup  is  a  solution 
of  .in  ZnSO4,  to  which  a  few  drops  of  Pb(NO3)2  are  added.  Calcu- 
late the  actual  concentration  of  the  metal  ions,  the  normal  concen- 
tration in  pure  water,  and  the  solubility  of  the  salt. 

QUESTIONS. 

1.  How  jnany  grams  of  the  salt  dissolve  at  this  temperature  in 
one  litre  of  water? 

2.  What  was  the  potential  difference  between  the  metal  and  the 
soluble  salt  solution?      (Table  XLIV  and  §3250.) 

In  the  last  experiment  we  neglected  the  difference  of  poten- 
tial between  the  two  different  electrolytes  which  were  sepa- 
rated by  the  porous  cup.  We  shall  now  consider  the  simplest 
case  of  such  a  junction. 


326         ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 


327.  Difference  of  Potential  Between  Two  Different  Solutions 
of  Equal  Concentrations.  —  If  the  velocities  of  the  ions  are 
unequal,  the  solution  with  the  faster  moving  cations  will 
thereby  lose  some  positive  electricity  which  will  be  gained  by 
the  other  solution.  The  same  effect  will  be  produced  if  the 
other  solution  has  the  faster  anions.  Planck1  has  calculated 
the  difference  of  potential  due  to  these  charges  and  finds 
that  for  two  binary  electrolytes,  of  like  ionic  concentration,  it 
is  equal  to 

r^  (2°3) 


FIG.  98. 

where  uI  and  «x'  are  the  velocities  of  the  cations  and  u2  and  u2 
are  the  velocities  of  the  anions.     The  reader  is  referred  to 
Planck's  paper  for  the  formal  proof  of  this  equation,  since 
only  a  quasi-proof  is  given  below. 

We  have  seen  previously  (Equation  202)  that  the  difference  of 
potential  between  two  identical  electrodes  in  solutions  containing 
common,  binary,  univalent  (v=i),  completely  dissociated  (1=2} 
ions,  whose  transport  number  is  approximately  .5,  is  given  by  the 
equation 

E=  1.98  X  10-4  0  log  — 

where  c2  and  d  are  the  concentrations  in  the  two  solutions. 

Let  Ui^>Uif  and  u2<^u2'.  Consider  a  portion  of  each  solution 
adjacent  to  the  boundary  (a  porous  wall  for  example,  see  Fig.  98). 

1  Wied.  Ann.,  1890,  xl,  p.  561. 


DISSOCIATION    OF    WATER.  327 

Owing  to  diffusion,  the  right-hand  solution  will  gain  an  excess 
of  positive  ions  which  is  directly  proportional  to  ult  the  velocity 
of  the  positive  ions  which  enter  and  u2'  the  velocity  of  the  negative 
ions  which  leave,  or  u^  +u2f.  This  gain  in  positive  ions  is  also 
evidently  inversely  proportional  to  the  velocities  of  the  positive 
ions  which  leave  the  right-hand  solution,  and  the  negative  ions  in 
the  other  solution,  or  u/  +u3.  Therefore,  the  ratio  of  concentra- 
tions of  positive  ions  is 


Ct        Uj  +  Uz 

This  same  expression  evidently  also  gives  the  proportion  in  which 
negative  ions  are  in  excess  on  the  left  side 

The  ionic  conductivities  (Table  XXXIX)  may  be  used  in 
place  of  the  actual  velocities  of  the  ions. 

328.  Dissociation  of  Water.  —  The  molecules  of  pure  water 
are  dissociated  to  a  slight  extent  (§258).  If  c=  concentration 
of  undissociated  water,  cx  =  concentration  of  hydrogen  ions, 
c2  =  concentration  of  hydroxyl  ions,  and  K  is  the  equilibrium 
or  dissociation  constant  (§251). 

CjCt  =  Kc 

K  may  be  determined  from  the  potential  difference  of  two 
hydrogen  electrodes,  one  of  which  is  in  a  known  acid  solution 
and  the  other  is  in  a  known  alkali  solution.  The  potential 
difference  will  give  the  concentration  of  H',  clf  in  the  alkali 
(§258)  and  the  concentration  of  OH/c2,  is  known,  c  is 

1000 

-  =  in 


In  pure  water 

The  values  of  K  at  different  temperatures  are  given  in  Table 
XLV. 

TABLE  XLV. 
Dissociation  of  Water.1 

Temperature o  25°  100°  200° 

Dissociation  constant io-is         io~l6        6XIO-1?         5Xio-18 

1  Heydweiller,  Ann.,  1909,  iii,  p.  503. 


328         ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

PROBLEMS  XXVII. 

1.  What  is  the  difference  of  potential  at  30°  between  two  hydrogen 
electrodes  dipping  in  a  common  electrolyte  but  under  pressures  of 
i  and  20  atmospheres,  respectively? 

2.  What  is  the  change  in  the  potential  at  20°,  between  a  metal 
and  a  solution  when  the  concentration  of  the  solution  is  increased 
tenfold,  if  (a)  the  metal  is  univalent  ?      (6)  if  it  is  bivalent  ?    (Assume 
N  =.5  and  a  =  i.) 

3.  What   is  the   potential   difference   at    10°   between   two   silver 
electrodes   which   dip   in   connecting   solutions   of   silver   nitrate,    if 
one  solution  contains  50  grams  of  salt  to  the  litre  and  the  other  2  ? 
(Obtain  i  from  Equations  112  and  187,  and   Table  XXXV,  and  N 
from    Table    XXXVIII    or    XXXIX.     The    dissociation    may     be 
considered  complete.) 

4.  What  is  the  difference  of  potential  at  22°   between    (a)  .00 in 
and  .000 1  w  solutions   of    silver   nitrate?     Similar   solutions  of    lead 
nitrate? 

5.  What  is  the  difference  of  potential  at  20°  between  two  silver 
plates,   one  of   which  is  in  a  .01   normal  solution  of    silver  nitrate 
and  the  other  is  in  a  saturated  solution  of  silver  bromide,  the  two 
solutions  being   separated   by   a  porous   cup  ?      (The   concentration 
of  the  silver  bromide -solution  may  be  taken  as  7  Xio-7.) 

6.  Calculate  the  concentration  of  OH  ions  at  20°  in  a    .1    normal 
solution  of  (a)  sulphuric  acid ;  (6)  acetic  acid.    [Calculate  first  the  con- 
centration of  H'  from  the  degree  of  dissociation  (Table  XXXV.)] 

7.  Calculate  the  concentration  of  hydrogen  ions  in  a  .in  solution 
of  (a)  potassium  hydroxide;  (6)  ammonium  hydroxide,  both  at  15°. 

8.  What  is  the  potential  difference  at  20°  between  .oin  solutions 
of  (a)  potassium  chloride  and  sodium  chloride  ?  (6)  potassium  nitrate 
and  sodium  chloride? 

9.  Calculate  the  difference  of  potential  at  25°  between  .in  and  .oin 
solutions  of  HC1. 

EXPERIMENT  LVI. 
Gas  Electrodes.     lonization  of  Water. 

Prepare  the  following  cell: 

H2 1  .oin  HC1 1  .oin  NaCl  |  .oin  NaOH  |  H2 

Each  hydrogen  electrode  consists  of  a  platinum  black  plate  sealed 
into  an  adapter  (see  Fig.  99). 

A  continuous  stream  of  hydrogen  bubbles  from  a  small  orifice,  D, 
below  the  plate,  and  escapes  through  a  second  small  tube  B.  E 
dips  in  the  intermediate  NaCl  solution.  One  electrode  contains  the 
acid  solution  and  the  other  the  alkali  solution. 

Determine  the  e.m.f.  of  this  cell  by  the  potentiometer  method 
(§73).  Observe  carefully  which  electrode  is  positive.  Calculate 
the  difference  of  potential  at  the  two  junctions  of  unlike  solutions 
(Equation  203  and  Table  XXXIX)  and  carefully  decide  whether 
these  potential  differences  are  in  the  same  direction  as  that  due 
to  the  electrodes  or  in  the  opposite  direction.  Substitute  the 
algebraic  difference  in  Equation  202  and  determine  the  concen- 
tration of  hydrogen  ions  in  the  alkali  solution.  The  concentrations 
of  hydrogen  ions  in  the  acid,  and  hydroxyl  ions  in  the  alkali,  may 


ENERGETICS    OF    A    CELL. 


329 


be  obtained  from  the  concentrations  and  the  degree  of  dissociation, 
calculated  from  their  equivalent  conductivities  as  given  in  Table 
XXXV.  Calculate  the  dissociation  constant,  K,  of  pure  water, 
the  concentration  of  hydrogen  and  hydroxyl  ions,  and  the  degree 
of  dissociation. 


FIG.  99. 


QUESTIONS. 

1.  Calculate   the   number   of   grams   of    (a)    hydrogen   ions?      (b) 
hydroxyl  ions  in  one  cubic  meter  of  pure  water. 

2.  What    percentage  error    in  your  result  would    be  caused  by 
neglect  of  the  potential  differences  between  the  solutions? 

3.  Why  must  both  electrodes  be  in  a  constant  stream  of  hydrogen? 

329.  Energetics  of  a  Cell.     Temperature  Coefficient. — We 

shall  now  consider  the  first  term  of  Equation  198  which  we 
have  eliminated  hitherto.  We  have  seen  in  §321  that  the 
mechanical  work  corresponding  to  the  second  term 

kRftn— 


may  be  expressed  in  terms  of  its  heat  equivalent.  Let  E!  be 
the  potential  of  one  electrode  of  a  cell  with  reference  to  its 
solution.  Let  Ql  be  the  heat  absorbed  during  the  passage  of 
16,530  mv  coulombs,  and  Q/  be  the  heat  equivalent  of  the 
energy  represented  by  the  accompanying  solution  or  deposition . 


330        ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

For  the  other  electrode 


The  total  electromotive  force  of  the  cell  is 

[(Q'-Q')  +  (G/-G*')]     (2°4) 


Let  E'  equal  a  similar  expression  for  the  electromotive  force 
at  a  lower  temperature  /'.  A  cycle  (§143)  between  these 
two  temperatures  will  eliminate  the  second  term  of  this 
equation.  For,  allow  the  cell  to  furnish  96,530?^  coulombs 
at  t°,  then  cool  it  to  tf  and,  from  some  outside  source,  send  the 
same  amount  of  electricity  -through  the  cell  in  the  opposite 
direction.  Finally,  restore  the  cell  to  its  initial  temperature 
t.  The  total  amount  of  chemical  change  is  zero,  and  there- 
fore, the  second  term  of  the  above  equation,  which  represents 
the  net  heat  value  of  the  chemical  reactions,  will  not  appear  in 
the  expression  for  the  energy  produced  by  this  cycle. 
The  net  work  done  by  the  cell  is 

W=  96530  mv  (E  —  Ef)  joules 

For  any  cycle,  the  work  W  and  the  heat  absorbed  at  the 
higher  temperature  (QI  —  Q2)  are  connected  by  the  relation 
(Equation  96) 

W_  -  6  ~  e'  -  96530  vw  (£-£') 

~Q=  0    'TI^TH^T 

.    n       n       96530™   E-E' 
'U'-y*-     4.187        0-0' 

.'  .  Equation  204  becomes 


ENERGETICS    OF    A    CELL.  331 

where  Qf  is  the  net  thermal  value  (calories)  of  the  reactions 
accompanying  the  passage  of  96,530  mv  coulombs, 

E-E'  =E-E'  =dE 
Q^¥  ==  /-  /'    :~  dt 

is  the  temperature  coefficient,  and  0  is  the  absolute  tempera- 
ture. If  Q'  represents  the  energy  of  the  reactions  in  calories 
for  the  deposition  or  liberation  of  one  gram  equivalent 

e^  (206) 

dt 

This  equation  is  often  called  the  GibbsI-Helmholtz2  equation 
from  the  names  of  the  discoverers.  If  the  second  term  is 
neglected,  it  is  often  known  as  Thomson's  rule. 

EXPERIMENT  LVII. 
Temperature  Coefficient  of  a  Cell.     Energetics  of  a  Cell. 

Prepare  either  of  the  following  cells: 

Ag  |  -  5*AgN03  1  ,  5nPb(NO3)2  1  Pb 
or 

Ag   .5ttAgNO3|.5wCu(NO3)2|Cu 

The  two  solutions  should  be  separated  bv  :\  porous  cup.  The  cell 
should  be  placed  in  a  thermostat  or  in  a  water-bath  whose  tempera- 
ture can  be  maintained  constant.  Determine  the  e.m.f.  (§73) 
at  two  different  temperatures.  Calculate  the  temperature  co- 
efficient 

dE      E-E' 


dt        t-t' 

Calculate  Q'  by  Equation  206  and  compare  it  with  Qi  —  QS  as 
calculated  for  one  gram  equivalent  from  the  data  given  in  Table 
XXVI. 

QUESTIONS. 

1.  Calculate  the  e.m.f.  of  the  cell  at  (a)  o°;  (6)  ipo°. 

2.  Would  this  cell  grow  colder  or  warmer  when  in  operation? 

3.  What  would  be  the  percentage  error  if  E  were  calculated  by 
Thomson's  rule? 

4.  Will  the  electrode  for  which  the  heat  of  solution  is  the  greater 
be    the    positive    or    the    negative    pole    for    external    connections? 
Explain. 

1  Trans.  Conn.  Acad.  1875,  iii,  pp.  108,  343. 

2  Berl.  Sitz.  Ber.  1882,  22. 


332         ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

Polarization. 

330.  To  send  a  current,  i,  through  an  electrolyte,  a  difference 
of  potential  E  must  be  applied  which  is  sufficient  to  overcome 
the  resistance  R  of  the  circuit  and  the  differences  of  potential 
E!  and  E2  at  the  electrodes 

E  =  iR  +  E>  +  E2     .-.i=E^E~E^  (207) 


is  called  the  polarization  electromotive  force.  El  and 
E2  have  the  values  given  in  Table  XLIV  (Le  Blanc's  law), 
and  may  be  either  positive  or  negative.  We  shall  consider  a 
few  cases. 

331.  Electrolysis  of  Copper  Sulphate.     Copper  Electrodes.— 
At  the  cathode  copper  is  deposited;  at  the  anode  copper  dis- 
solves.    Evidently  E^  =  —E2,  or  the  polarization  electromotive 
force,  is  zero.    "If,   however,  the  e.m.f.  is  so  great  that  the 
copper  ions  are  insufficient  to  carry  all  the  current,  hydrogen 
(from  the  dissociation  of  water,   §§258  and  328)  will  also  be 
liberated  at   the   cathode.     Table   XLIV   shows  that   while 
copper  is  so   desirous  of  depositing  from  a  normal   copper 
solution  that  it  continues  to  do  so  until  the  potential  reaches 
.606  volts,  hydrogen,  in  normal  concentration,  is  not  spon- 
taneously deposited  at  the  cathode  after  the  potential  reaches 
.277  volts.     At  the  anode  the  potential  of  .606  volts  must 
be  overcome  if  copper  is  to  dissolve,  therefore  the  polarization 
e.m.f.  is  .329  if  hydrogen  from  a  normal  solution  is  liberated 
with  the  copper.     The  actual  concentration  of  the  hydrogen  is 
much   less  and  therefore  (§3250)   the   polarization   e.m.f.   is 
greater. 

332.  Platinum   Electrodes.  —  The   cathode   is   soon   covered 
with  copper,  and,  being  indistinguishable  from  a  copper  plate, 
shows  a  potential  difference  of  .606  volts  above  the  solution 
(Le  Blanc's  law).     If  the  current  is  weak,  the  oxygen  ions 
can  convey  it  to  the  anode.     1-396  volts  would  be  required 
(see  note*at  end  of  §334)  and  the  minimum  polarization  e.m.f. 
would  be  .79  volts.     If  the  current  is  gradually  increased,  OH' 
jons,  SO4"  ions,  and  possibly  HSO/  ions  will  also  be  liberated 


ELECTROLYSIS  OF  SULPHURIC  ACID.         333 

and  Table  XLIV  shows  that  these  will  give  greater  polariza- 
tion e.m.f.'s.  Since  these  radicals  are  liberated  in  the  presence 
of  water,  they  will  change  to  equivalents  of  oxygen  by  obvious 
reactions,  before  they  leave  the  solution. 

333.  Electrolysis    of    Sulphuric    Acid    Between    Platinum 
Plates. — Suppose  we  have  normal  ionic  concentration.     The 
liberation  at  the  anode  will  be  identical  with  that  described 
in  the  preceding  paragraph.     Hydrogen  is  liberated  at    the 
cathode    giving   a   difference   of    potential   of    +  .277.      Ac- 
cording to  the  magnitude  of  the  current,   the   polarization 
e.m.f.  will  therefore  be  about 

1.396  —  .277  =1.119  volts,  1.94  — .28  =1.66  volts, 
2.2— .28  =  1.92  volts,  or  2.9— .28=2.62  volts. 

There  is  little  evolution  of  gas  until  the  OH'  ions  are  liberated. 

334.  Electrolysis  of  Sodium  Hydroxide  Between  Platinum 
Electrodes. — Oxygen  will  be  liberated  at  the  anode  if  the 
current  is  extremely  small,  otherwise  hydroxyl  ions  will  also 
be  liberated  and  about   1.94  volts  must  be  overcome.     At 
the  cathode  there  are  both  the  sodium  ions  and  the  hydrogen 
ions  from  the  dissociation  of  the  water.     The  hydrogen  ions, 
however,  will  be  liberated  unless  the  current  is  very  great  and 
consequently  the  fall  of  potential  is  very  high,  for  normal 
hydrogen  ions  give  a  potential  of  .277  which  assists  the  elec- 
trolyzing  current  making  the  polarization  e.m.f.    1.66  volts, 
while  a  cathode  fall  of  potential  of  two  volts  is  necessary  to 
liberate  sodium, making  the  total  polarization  e.m.f.  2.28  volts. 

In  all  these  examples,  values  of  the  potential  have  been 
used  which  are  only  correct  for  one  particular  concentration 
and  therefore  the  results  are  only  approximate.  If  the 
concentrations  are  known,  the  true  potential  differences  may 
be  calculated  from  Table  XLIV  and  §3250. 

EXPERIMENT  LVIII. 
Polarization. 

Pour  the  assigned  electrolyte  into  a  large  crystallizing  dish, 
mount  in  it  two  platinum  electrodes,  and  connect  them  to  a  source 
of  current  through  a  regulating  rheostat,  a  switch,  and,  preferably, 


334         ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 


an  ammeter  (see  Fig.  100).  Arrange  also  a  compensation  apparatus 
and  a  normal  calomel  electrode  (§§73,  75)  so  as  to  determine  the 
potential  difference  between  each  electrode  in  succession,  and  the 
solution  at  its  surface.  Determine  also  the  total  fall  of  potential 
between  the  electrodes.  Repeat  for  several  other  current  strengths. 
Calculate  the  potential  drop  at  each  electrode,  the  total  e.m.f. , 
and  the  polarization  e.m.f.  Careful  attention  must  be  paid  to  signs 
(§76). 


^AAA/WVWV\AMAA/VW 

r    


Rheostat 

AAMM/WV 


Storage 
Battery 


FIG.  100. 


QUESTIONS. 

1.  What   was  the  fall   of  potential    (a)    between  the   electrodes? 
(6)  in  the  electrolyte?      (c)  the  resistance  of  the  electrolyte? 

2.  What  mass  of  metal  would  have  been  deposited  on  the  cathode 
in  one  hour? 

3.  Does  the  fall  of  potential  at  the  anode  help  or  hinder  the  cur- 
ent?  at  the  cathode? 


335- 


Galvanic  Cells. 

The  earliest  arrangement  for   producing  an   electric 


current  from  the  difference  of  potential  of  two  dissimilar 
metals,  dipping  in  an  electrolyte,  consisted  of  a  copper  and 
a  zinc  plate  dipping  in  dilute  sulphuric  acid.  When  the  metals 


STANDARD    CELLS.  335 

were  joined,  a  current  flowed  through  the  circuit,  the  direction 
in  the  solution  being  from  zinc  to  copper.  The  copper  plate 
soon  became  covered  with  the  readily  liberated  hydrogen 
which  lowered  the  potential  of  the  cell  from  about  .493  +.606 
to  about  .493  +.277  (Table  XLIV)  and  interposed  great  resis- 
tance to  the  passage  of  the  current. 

Various  methods  were  devised  to  remove  this  hydrogen,  and 
the  cells  which  have  accomplished  this,  without  introducing 
other  serious  faults,  will  be  described  below. 

336.  (A)  Cadmium  Cell  (Reichsanstalt  Form).     Cadmium 
amalgam — CdSO4  crystals— CdSO4  solution— Hg2SO4  paste — 
amalgamated  platinum  + 

The  chemical  reactions  consist  of  solution  of  cadmium  and 
deposition  of  mercury. 

Et  —  1.01843  ~~  .00004075  (t—  20)  —  .00000095  (^~  2o)2*  (208) 

337.  (B)  Cadmium    Cell    (Weston    Form). — The  cadmium 
sulphate  crystals  are  omitted.     Its  e.m.f.  may  be  taken  as 
1.0190  volts. f 

338.  (C)  Clark  Cell.— 

Zn  -  ZnSO4  •  7  H2O  crystals  - ZnSO4  solution  - Hg2SO4  -  Hg  + 

Solution  of  zinc,  deposition  of  mercury. 
e.m.f.  =  1.4328  —  .00119  (/—  15°)  —  .000007    (^~I5°)2t    (2°9) 

339.  (D)  DaniellCell. 

Zn  -  ZnSO4  •  x  H2O  |CuSO4  •  x  H2O  -  Cu  + 

Solution  of  zinc,  deposition  of  copper.  The  division  A 
may  be  a  porous  cup.  or,  since  the  CuSO4  solution  is  heavier, 
the  two  solutions  may  be  separated  by  gravity. 

E.m.f.  =1.105  volts  (approximate).  The  internal  resist- 
ance is  generally  high  (i-io  ohms) . 

*Jager  and  Steinwehr,  Zeit.  Inst.,  1908,  p.  327;  Wolff,  Bui.  Bu.  Stand.,  1908, 

P-  309- 

f  Winkelmann,  1905,  iv,  I,  p.  208. 
tjager  and  Kahle,  Ann.,   1898,  Ixv,  p.  926. 


336        ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

(A),  (B)  and  (C)  give  very  constant  electromotive  forces 
when  the  current  is  inappreciable.1  A  and  B  have  the  advan- 
tage of  having  a  very  small  temperature  coefficient.  All  three 
have  very  high  internal  resistances  and  cannot  be  used  to 
supply  current. 

The  e.m.f.  of  (D)  is  sufficiently  constant  for  ordinary 
laboratory  use  even  if  the  cell  is  supplying  an  appreciable 
current . 

340.  (E)  Edison  Cell. 

Zn  -  KOH  -  Cu2O  + 

Solution  of  zinc,  reduction  of  copper  oxide, 
e.m.f.  =  8.     Very  low  internal  resistance  (e.g.,  about  .05  ohm). 

341.  (F)  Leclanche    Cell. — Zn —concentrated   NH4C1   solu- 
tion —  (MnO2  +  C)  -f .     Solution  of  zinc,  reduction  of  manganese 
dioxide. 

e.m.f.  =  1.46      Moderate  internal  resistance  (e.g.,  .5  ohm). 

342.  The  dry  cell  is  a  Leclanche  cell  in  which  the  liquid  is 
absorbed  by  plaster  of  Paris  or  some  similar  absorptive.     The 
Leclanche*  cell  is  cheap  and  very  satisfactory  when  a  current  is 
only  required  for  short  times,  at  infrequent  intervals. 

343.  Lead  Storage  Cell. 

Pb  -  H2S04  •  x  H20  (density  =1.2)-  PbO2  + 

The  reaction  at  the  positive  pole  (the  cathode  as  regards 
the  passage  of  the  current  through  the  electrolyte  during  dis- 
charge) is  (§287) 

Pb02+  2H-  +  28  +  HaS045F»PbS04  +  2H2O 
and  the  reaction  at  the  other  pole  is 

Pb  +  SO/  =  PbSO4+  20 

E.m.f.  =  2.0  to  2.2  volts.  Very  low  internal  resistance. 
The  weight  of  lead  storage  cells  is  about  10  kilos  per  100  watt 
hours. 

1  Directions  for  setting  up  these  cells  are  given  by  Wolff  and  Waters,  Bui. 
Bu.  Stand.,  1907,  p.  623.  Also  Watson,  "Practical  Physics,"  p.  487. 


EDISON    STORAGE    CELL.  337 

As  the  storage  cell  discharges,  both  >plates  change  to  lead 
sulphate  and  the  concentration  of  sulphuric  acid  decreases. 
To  charge  the  cell  a  current  is  sent  through  in  the  opposite 
direction  and  the  reactions  are  reversed.  Evidently  the 
positive  terminal  when  the  cell  is  discharging  is  the  terminal 
to  which  the  positive  side  of  the  supply  current  should  be 
connected.  The  positive  pole  is  obviously  an  electrode  of 
the  second  class  (§31  9).* 

344.  Edison  Storage  Cell.— 

(Fe  +  C)  -  20%  KOH  -  (Ni(OH)3  +  C) 

E.m.f.  =  1.2.  The  internal  resistance  is  even  less  than 
that  of  the  lead  storage  cell.  Other  advantages  are;  a  more 
constant  e.m.f.,  it  may  be  both  charged  and  discharged  at 
much  higher  currents,  and  the  weight  per  100  watt  hours  is 
only  about  3.5  kilos.2 

PROBLEMS  XXVIII. 

1.  Derive    Equation    206    directly   from    Equation    101.      (Notice 
that   7=0'    (§321)    and    fP-£*-o6c«o£.) 

2.  Derive   Equation   206   directly  from   Equations    172    and    173. 
(Notice  that  W  =Ee.     Find  the  complete  differential  of  Equation  172, 
substitute  from  Equation  173,  and  reduce.) 

3.  Calculate  the  electromotive  force  of    a   Daniell  cell    from  the 
heat    values    of    the    chemical    reactions.      (Table    XXVI.)      What 
error  has  the  result  ? 

4.  What  proportion  of  the  energy  of  a  lead  storage  cell  comes 
from  the  energy  of  the  chemical  reactions? 


[Pb  +  PbO2  +  2  H2SO4+aq  =  2PbSO4+2H2O  +  aq  +  85ooo  calories 
(Van  Laar)] 

5.  Explain  the  liberation  of  hydrogen  when  a  storage  cell  is  being 
overcharged. 

6.  Two   platinum   electrodes   dip   in   a   normal   solution   of  silver 
nitrate.     Calculate    the    different    polarization    electromotive    force 
as  the  current  is  gradually  increased. 

7.  Is   the  mercury  electrode  of   the    cadmium    cell    an    electrode 
of  the  first  or  second  class?     Of  the  Clark  cell? 

8.  If  the  solubility  of  cadmium  sulphate  changes  with  the  tempera- 
ture, which  form  of  cadmium  cell  will  have  the  smaller  temperature 
coefficient?     Explain. 

'For  further  information  consult  Dolezalek,  "Theory  of  the  Lead  Accumu- 
lator," Wiley,  New  York. 

2For  further  information  consult  Foerster,  "  Electrochemie,  "  pp.  153-159. 


338         ELECTROCHEMISTRY.       POTENTIAL    DIFFERENCES. 

9.  What  changes  would  be  produced  in  the  e.m.f.   of  a  normal 
Daniell  cell   if    (a)    the    concentration   of  the    copper  sulphate   was 
reduced  to  one-tenth  of  its  value?      (6)  If  the  concentration  of  the 
zinc  sulphate  was  reduced  instead? 

10.  If  zinc  costs  15  cents  per  pound,  copper  oxide  40  cents,  copper 
20  cents,  caustic  potash  15  cents  a  pound,  calculate  the  approximate 
cost  of  one-horse-power  hour  obtained  from  Edison  cells.      (Neglect 
the  value  of  the  zinc  salt  formed.) 


EXPERIMENT  LIX. 
Study  of  a  Storage  Cell. 

Observe  the  current,  voltage,  and  time  as  a  storage  cell  is  charged 
and  also  as  it  is  discharged  through  a  known  resistance.  The 
current  should  not  exceed  .01  ampere  per  cm.2  of  either  electrode, 
and  if  the  cell  is  valuable,  the  discharge  should  not  be  continued 
after  the  e.m.f.  of  the  cell  falls  below  1.5  volts. 

Plot  all  the  observations  against  the  time. 

If  two  channeled  or  pocketed  lead  plates  are  available,  it  is 
interesting  and  instructive  to  prepare  the  cell-  which  is  tested.  The 
process  of  forming  the  plates  .is  greatly  hastened  if  the  positive 
plates  are  filled  with  red  lead,  Pb3  O4,  and  the  negative  with  litharge, 
PbO.  These  oxides  should  be  prepared  as  thick  pastes,  by  mixing 
thoroughly  with  dilute  H2SO4  (s.  g.  =  1.2)..  When  all  reaction  has 
ceased,  the  pastes  should  be  pressed  into  the  carefully  cleaned  plate 
pockets.  After  having  dried  for  at  least  one  day,  they  are  placed 
in  a  suitable  jar  filled  with  dilute  H2SO4  (s.  g.  =  1.2).  The  positive 
and  negative  plates  must  be  carefully  insulated. 


QUESTIONS. 

1.  Calculate    (a)    the  initial    and    (6)   the    final    activity    (watts) 
of  the  cell  during  charging  and  discharging. 

2.  Why  are  the  curves  for  charging  and  discharging  dissimilar? 

3.  Calculate  by  Faraday's  law,  the  amount  of  lead  sulphate  which 
disappeared  during  charging. 


CHAPTER  X. 
GASEOUS  IONS.     RADIOACTIVITY. 

345.  Conduction  of  Electricity  Through  Gases. — When  gases 
are  subjected  to  radioactive  bodies,  X-rays,  ultra-violet  light, 
hot  metals,  or  flames,  or  intense  electric  forces,  the  molecules 
break  up  into  ions.  These  ions  resemble  electrolytic  ions 
(§§284,  285)  in  that  they  have  a  charge  which  is  usually 
equal  to  that  on  either  a  univalent  or  bivalent  electrolytic 
ion.  At  very  low  pressures  the  positive  ions  are  usually  atoms 
or  molecules  with  a  univalent  charge,  while  the  negative  ion 
is  a  so-called  "electron"  with  a  univalent  electrolytic  charge 
(§285)  and  a  mass  of  about  1 7  70  Xio8 -1-96530  =T¥1^TFthat  of  an 
hydrogen  atom.1  At  ordinary  pressures,  these  ions  become 
the  nuclei  of  clusters  of  several  molecules. 

This  very  brief  introduction  is  preparatory  to  the  considera- 
tion of  the  equilibrium  of  certain  endothermic  compounds 
under  particular  forms  of  electric  discharge. 

Ozone  and  the  oxides  of  nitrogen  are  endothermic  com- 
pounds, and  therefore  (§191)  the  stable  concentration  increases 
with  the  temperature.  At  ordinary  temperatures,  under 
natural  conditions,  it  is  negligible,  but  under  the  influence  of 
a  silent,  luminous,  electric  discharge  the  natural  constructive 
forces  are  strengthened  and  the  decomposing  forces  are  re- 
duced, so  that  the  equilibrium  concentration  is  enormously 
increased,  particularly  at  low  temperatures. 

If  oxygen,  for  example,  is  subjected  to  such  a  silent  dis- 
charge, a  considerable  portion  may  be  transformed  into  ozone. 
The  ozone  will  only  slowly  return  to  the  negligible  equilib- 
rium concentration  when  removed  from  the  electric  field,  for 
at  moderate  temperatures  the  decomposing  forces  are  very 
weak  (although  much  greater  than  the  constructive,  if  the 
concentration  is  large) . 

1  Classen,  Phys.  Zeit.,  1908,  ix,  p.  762. 

339 


340 


GASEOUS    IONS.       RADIOACTIVITY. 


346.  The  silent  electric  discharge  is  an  extensive,  quiet,  low 
temperature    conduction    of  electricity  through  a  gas.     The 
gas  through  which  the  electricity  is  passing  has  a  violet  hue. 
It  differs  from  the  spark  discharge  in  that  the  latter  is  confined 
to  a  very  narrow  path  through  the  gas,  while  the  silent  dis- 
charge always  occupies  an  appreciable  volume  of  gas  and  in 
commercial  ozonizers  it  often  involves  a  large  volume.     If 
the  potential  is  not  excessive,  electricity  will  pass  between  a 
point  (preferably  positive)  and  a  plate  by  the  silent  discharge. 
If  a  high,  alternating,  electromotive  force  is  applied  to  two 
conductors  which  are  separated  by  the  gas,  and  a  dielectric 
plate  (for  example,  glass)  is  interposed  anywhere  between  the 
two  conductors,  the  gas  may  be  subjected  to  a  sufficient  electric 
force  to  produce  the  ions  which  carry  the  silent  discharge,  and 
the  dielectric  will  prevent  sparks. 

347.  Production  of  Ozone. — Table  XLVI  gives  the  efficiency 
of  different  methods  of  producing  ozone.     The  thermal  method 
depends  upon  the  principles  of  §191  and   §269,  and  has  been 

TABLE  XLVI. 
Grams  of  Ozone  per  Kilowatt  Hour  by  Different  Processes.1 


Concentration 
(gr.  per  cu.  m.) 

Thermal  !  Thermal 
(blast)      (liq.  air) 

Electrolysis 

Silent  Discharge 

Points 

Dielectric 

•37 
1.8 

14 
290 

i.-3 

3-5 

2 

7 

+ 
60 
5° 

40 

37 

10 

70 

55 

fully  worked  out  by  Franz  Fischer.2  The  electrolytic  method 
depends  upon  the  fact  that  in  the  electrolysis  of  acidulated 
water,  ozone  as  well  as  oxygen  is  liberated  at  the  anode,  par- 
ticularly if  the  current  density  is  very  great. 

348.  Production  of  Oxides  of  Nitrogen. — The  silent  discharge 

IEwell,  "Electrochemical  and  Metallurgical  Industry,"  1909,  i,  p.  23. 
2Berichte  deutsch.  Chem.  Gesell.  xxxix,  pp.  940,   2557,  3631;  xl,  pp.  443, 
ii n;  xli,  p.  945. 


RADIOACTIVE    BODIES.  341 

through  air,  at  atmospheric  pressure,  at  20°,  oxidizes  about 
10  litres  of  nitrogen  per  ampere-hour.1  The  most  efficient 
means  of  oxidizing  the  nitrogen  in  the  air  is  to  raise  it  to  a  very 
high  temperature  by  means  of  an  electric  arc  and  then  quickly 
remove  it  from  the  hot  region  by  giving  both  the  air  and  the 
arc  a  rapid  motion  at  right  angles.  The  large  commercial 
plants  in  Norway  which  use  the  Birkeland  process  yield  about 
1.5%  to  2%  of  nitrogen  oxides  which,  when  absorbed  by  water 
give  1 20  grams  of  50%  pure  nitric  acid  per  kilowatt  hour.2 

A  cheaper  method  of  obtaining  the  same  amount  of  nitrogen 
in  a  suitable  form  for  fertilizers  is  to  absorb  the  pure  nitrogen 
produced  by  the  Claude  or  Linde  process  (§§138,  139)  in 
calcium  carbide.  Calcium  cyanamide  is  formed  according  to 
the  reaction 

CaC2  +  N2  =  CaCN2  +  C 

One  kilowatt  hour  yields  130  gr.  of  CaC2  and  about  1000  gr. 
of  N2,  or  about  180  gr.  of  CaCN2.3 

349.  Radioactive  Bodies. — The  rays  emitted  by  radio- 
active bodies  as  they  disintegrate  consist  of  electrons  at  enor- 
mous velocities  (some  have  nearly  the  velocity  of  light), 
helium  atoms  with  the  charge  on  a  bivalent  electrolytic  ion  4 
and  a  velocity  which  may  be  as  high  as  one-fifteenth  that  of 
light,  and  X-rays.  These  rays  are  often  designated  as  /?,  a, 
and  j-  rays,  respectively. 

The  a  rays,  or  helium  atoms,  are  absorbed  by  a  few  centi- 
meters of  air,  or  a  few  hundredths  of  a  millimeter  of  aluminum. 
Therefore,  a  large  part  of  the  a  rays  emitted  in  the  interior  of 
a  radioactive  body  never  reach  the  surface.  The  mass  of  a 
rays  is  so  much  greater  than  the  mass  of  the  /?  rays  that  they 
have  much  more  energy,  and  when  they  are  stopped  they  give 
up  this  energy  in  the  form  of  heat  and  light.  One  gram  of 
radium  develops  about  100  calories  per  hour. 

The  /?  rays  (electrons)  have  the  mass  stated  in  §3 45  (1/1830) 

1  Warburg  and  Leithauser,  Ann.,  1906,  xx,  p.  743. 

2Haber,  "Thermodynamics,"  p.  267. 

3  Elec.  Chem.  Ind.,  1907,  pp.  77,  491. 

4 Rutherford,  Proc.  Roy.  Soc.,  1908,  A,  81,  p.  162. 


342  GASEOUS    IONS.       RADIOACTIVITY. 

that  of  an  hydrogen  atom  —  provided  the  velocity  is  far  re- 
moved from  the  velocity  of  light  (3  X  io10  cm/  sec)  . 

350.  Variation  of  Mass  with  Velocity.  —  If  the  velocity  is 
increased  above  about  one-tenth  the  velocity  of  light,  the 
mass,  as  defined  by  Newton's  second  law,  increases  also.  If 
m0  =mass  for  moderate  velocities,  mu  =mass  when  the  velocity 
has  a  high  value,  u,  and  U  =  velocity  of  light. 

£=  (2IO) 

"' 


If  this  equation  represents  the  mass  of  any  body  of  matter 
rather  than  simply  that  of  an  electron  (and  there  are  good 
grounds  for  believing  that  such  is  the  case)  the  mass  of  a  body 
cannot  be  regarded  as  absolutely  constant.  However,  u/U 
is  inappreciable  in  terrestrial  mechanics  and  for  all  practical 
purposes  the  mass  of  a  body  may  still  be  regarded  as  constant. 
The  student  is  urged  to  read  Lewis'  derivation  of  this  equation 
and  discussion  of  its  consequences.1 

351.  Table  XL  VI  I  illustrates  the  series  of  changes  of  uranium, 
radium,  etc.,  and  the  values  of  the  time  constant  or  period 
(i.e.,  the  time  required  for  the  subtance  to  be  half-trans- 
formed). The  time  constant  is  determined  from  the  velocity 
constant  as  described  in  §238.  The  calculation  is  often  com- 
plicated by  the  successive  reactions. 

The  emanation  is  a  gas  under  ordinary  conditions.  Ruther- 
ford found  that  its  boiling  point  is  —68°  under  atmospheric 
pressure  and  that  its  molecular  weight  is  222.  By  compressing 
the  emanation  into  a  capillary  tube  and  cooling  it  with  liquid 
air,  Rutherford  obtained  visible  amounts  of  liquid  which 
glowed  with  a  brilliant  phosphorescence.2  Notice  that  the 
difference  between  the  atomic  weights  of  radium  (226)  and  that 
of  the  emanation  is  the  atomic  weight  of  the  expelled  a 
particle  -(helium).  Sufficient  helium  accompanies  the  disin- 
tegration of  the  emanation  to  be  detected  spectroscopically 

TPhil.  Mag.,  xvi,  p.  705 

2Phil.  Mag.,  1909,  xvii,  p.  723. 


DISINTEGRATION    PRODUCTS    OF    URANIUM. 


343 


within  one  day.1  The  product  formed,  radium  A,  collects 
on  the  walls  of  the  containing  vessel,  but  if  any  body  in  the 
vessel  has  a  negative  charge,  all  of  the  radium  A  will  collect 
upon  it.  This  phenomenon  is  known  as  excited  activity. 

TABLE  XL VII. 

Disintegration  Products  of  Uranium.2 


Radioactive  Products. 


Period 


Nature  of  Rays 
Emitted. 


Uranium 5  X 100  years  a 

I 
Uranium  X j  22  days  /?  and 

l 
Ionium :  ?  a 

* 

Radium 2,000  years  a 

4- 
Emanation  375  days  a 

I. 

Radium  A 3  minutes  a 

i 

Radium   B 26  minutes  a 

* 
Radium   C j        19  minutes  a,  /?,  and  f 

I 
Radium  D    (radio-lead) |         40  years  No  rays 

Radium  E 6  days  No  rays 

I 
Radium  F 45  days  /?  and  f 

Radium  G  (polonium) 140  days  a 

i 


Radioactive  changes  are  best  followed  by  studying  the 
conductivity  of  gases  produced  by  the  three  types  of  rays. 
They  may  also  be  studied  by  their  effect  on  a  photographic 
plate  and  by  the  fluorescence  which  they  produce.  Most 
of  the  elements  are  probably  disintegrating,  but  very 
much  slower  than  radium,  thorium,  actinium,  etc.  For 
further  information  respecting  the  conduction  of  electricity 
through  gases,  and  radioactivity,  see  the  treatises  of  Thomson, 
Rutherford,  Strutt  and  McClung. 

Rutherford,  Phil.  Mag.,  1909,  xvii,  p.  281. 
'McClung,  in  "A  Text-book  of  Physics,"  Duff. 


344 


GASEOUS    IONS.       RADIOACTIVITY. 
PROBLEMS  XXIX. 


1 .  It  is  desired  to  produce  one  kilo  of  ozone  of  such  a  concentration 
that  1.8  gr.  are  contained  in  i  cubic  meter,      (a)  How  many  calories 
are  required  by  the  thermal  process?      (6)   How  long  a  time  would 
a  one-horse-power  dielectric  ozonizer  require? 

2.  Calculate  the  amount   of    oxides  of    nitrogen  produced  by   a 
looo-horse-power  plant  in  one  year  if  the  plant  is  in  operation  300 
complete  days. 


Air+0, 


\ 


FIG.  101. 

3.  Assuming  the  validity  of  Equation  210,  calculate  the  true  mass 
of  a  20  gram  bullet  traveling  with  a  velocity  of  1000  m.  per  sec. 

4.  A  vessel   filled  with  radium   emanation  showed   the   following 
activity  (in  arbitrary  units)  at  different  times. 

t  (hours)  Activity 

o  100 

20.8  85.7 

187.6  24 

355-  6.9 

522-  i-S 

787.  .19 


ELECTRICAL    PRODUCTION    OF    OZONE. 


345 


Calculate  (a)  the  velocity  constant,  (6)  the  time  in  which  half 
the  emanation  is  transformed  (Meyer,  p.  25). 

EXPERIMENT  LX. 
Electrical  Production  of  Ozone.1 

For  a  description  of  more  efficient  apparatus  and  for  extensive 
general  details,  see  the  references  below.  Siemens  (dielectric) 
ozonizers  are  sold  by  glass  instrument  makers  under  the  name  of 
ozone  tubes  (see  Fig.  101).  A  point  ozonizer  may  be  constructed 
by  sealing  several  platinum  wires  into  a  large  glass  tube  and  mount- 
ing opposite  them  a  platinum  plate  (see  Fig.  102). 

This  latter  type  of  ozonizer  should  be  connected  to  a  plate  electric 
machine.  The  former  (dielectric)  type  requires  a  suitable  induction 
coil  or  a  suitable  transformer.  The  latter  is  preferable  except  for 


Air 


Air  t  03 


FIG.  102. 

the  danger  from  serious  shocks.  Connect  a  series  of  drying  tubes 
and  a  tube  filled  with  glass  wool  to  the  side  of  the  ozonizer  at  which 
the  air  or  oxygen  enters.  Connect  to  the  other  end,  two  gas  absorption 
vessels,  preferably  of  the  Gray  type,2  one  of  which  contains  water  to 
absorb  oxides  of  nitrogen,  and  the  other  contains  .in  KI  solution. 
The  fall  of  potential  in  the  tube  is  determined  by  a  Braun  electrom- 
eter which  is  connected  across  the  terminals.  A  non-inductive 
resistance,  for  example,  a  potassium  iodide  solution,  or  a  non- 
inductive  coil,  is  put  in  series  with  one  of  the  terminals.  If  a  point 
tube  and  a  plate  machine  are  used,  a  galvanometer  is  connected 

1  Warburg,  Ann.,  ix,  pp.  781,  1287;  xiii,  p.  464;  xvii,  p.  i;  xx,  p.  134;  xxiii, 
p.  209;  xxviii,  pp.  i,  17.  Ewell,  Phys.  Rev.,  1906,  xxii,  pp.  3,  232;  Am.  Jo. 
of  Science,  1906,  xxii,  p.  368;  Phys.  Zeit.,  1906,  xxv,  p.  927;  Elec.  Chem.  Ind., 
1907,  p.  264;  1909,  p.  23. 

2Ann.    1904,  xiii,  p.  466. 


346  GASEOUS    IONS.       RADIOACTIVITY. 

across  the  non-inductive  resistance,  and  a  Dolezalek  electrometer 
(idiostatically  connected)  is  used,  if  a  dielectric  ozonizer  is  employed. 
The  instrument  and  non-inductive  resistance  are  calibrated  together, 
as  an  ammeter  (§78). 

The  current  is  adjusted  until  the  proper  purple  silent  discharge 
is  obtained.  The  time  is  then  noted  and  a  steady  current  of  air  or 
oxygen  is  forced  or  drawn  through  the  apparatus.  The  velocity 
of  the  gas  is  determined  either  with  a  gasometer  or  by  collecting 
a  certain  volume  over  water  and  noting  the  time  required.  When 
sufficient  ozone  has  been  produced  and  absorbed,  the  time  is  noted 
and  the  current  stopped,  and  a  few  moments  later  the  gas  current 
is  also  shut  off.  The  amount  of  ozone  is  determined  by  acidulating 
the  .in  KI  solution  with  an  equivalent  of  H2SO4,  adding  starch 
and  titrating  with  .O2n  Na2S2O3.  i  c.c.  of  the  latter  =.48  mg.  of 
ozone. 

Calculate  (a)  total  yield  of  ozone ;  (b)  yield  in  grams  per  coulomb ; 
(c)  yield  in  grams  per  kilowatt  hour;  (d)  concentration  in  grams 
per  cubic  meter. 

QUESTIONS. 

1.  Calculate  the  amount  of  ozone  which  would  have  been  liberated 
by  the  quantity  of  electricity  which  traversed  the  tube  if  the  process 
were  electrolytic. 

2.  What  electrical  reason  is  there  for  carefully  drying  the  gas? 
(Other  objections  to  moisture  are  given  in  Warburg's  paper,  Ann.,  1906,  xx, 

P-  751-) 


TABLES. 

All  the  tables  in  the  book,  whether  they  appear  here  or 
earlier,  are  listed  in  the  following  pages  under  their  respective 
subjects.1 

Mathematical  Tables,  Densities,  etc. 

Symbols,  (I)  p.  i. 


1  For  more  detailed  information,  consult  the  tables  of  Landolt  and   Bornstein, 
the  Smithsonian  Tables,  and  Winkelmann's  Handbuch. 


348 


TABLES. 


TABLE   XL VIII. 
Logarithms  of  Numbers  from  i  to   1000. 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

O2I2 

02  53 

0294 

0334 

0374 

ii 

0414 

°4<3 

0492 

Q531 

0569 

0607 

0645 

0682 

0719 

°755 

12 

0792 

0828 

0864 

0899 

°934 

0969 

1004 

1038 

1072 

1106 

13 

IJ39 

JI73 

1206 

1239 

1271 

!3°3 

J335 

J3^7 

J399 

143° 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

!673 

J7°3 

J732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

J931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

233° 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

262  5 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3*39 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

354i 

356o 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

'3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

399? 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4i33 

26 

415° 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

43X4 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

2Q 

4624 

4639 

4654 

4669 

4683 

4698 

47*3 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

501  1 

5024 

5038 

32 

5051 

5065 

5°79 

5092 

5!°5 

5H9 

5132 

5r45 

5159 

S1?2 

33 

5185 

5*98 

5211 

5224 

5237 

525o 

5263 

5276 

5289 

5302 

34 

53i5 

5328 

5340 

5353 

5366 

5378 

539i 

5403 

54i6 

5428 

35 

544i 

5453 

5465 

5478 

549° 

5502 

5515 

5527 

5539 

555i 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

57°5 

57J7 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

59U 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

655J 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

691  1 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

5i 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7J35 

7J43 

7I52 

52 

7  1  60 

7168 

7i77 

7185 

7J93 

7202 

72  10 

7218 

7226 

7235 

53 

7243 

725i 

7259 

7267 

7275 

7284 

7292 

7300 

73o8 

73J6 

54 

7324 

7332 

7340  | 

7348 

7356 

7364 

7372 

738o 

7388 

7396 

No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

TABLES. 

TABLE   XLVIIL— Continued. 
Logarithms  of  Numbers  from  i  to  1000. 


349 


No. 

1  « 

i 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

75*3 

7520 

:  7528 

7536 

7543 

7551 

i  *j  *j 

57 

7559 

7566 

7574 

7582 

7589 

7597 

i  7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

1  7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

•  7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8i95 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

7i 

8513 

8519 

8525 

853i 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

859i 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9oi5 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9*33 

82 

9*38 

9M3 

9149 

9!54 

9i59 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

93°4 

93°9 

93T5 

9320 

9325 

933° 

9335 

9340 

86 

9345 

935° 

9355 

9360 

9365 

937° 

9375 

938o 

9385 

939° 

87 

9395 

9400 

9405 

9410 

94i5 

9420 

9425 

943° 

9435 

9440 

88 

9445 

945° 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

95r3 

95i8 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

958i 

9586 

9i 

959° 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

97J3 

9717 

9722 

9727 

94 

9731 

9736 

974i 

9745 

975° 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

993° 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

No. 

0 

i 

2 

4 

5 

6 

* 

8 

9 

350 


TABLES. 


TABLE   XLIX. 
Natural  Sines  and  Cosines. 


Sine 

£>i° 

1 

Cosine 

£>i° 

1  o 

o.oooo 

90 

I.OOOO 

I 

0.0175 

i75 

89 

0.9998 

02 

2 

0.0349 

J74 

88 

0.9994 

04 

3 

0.0523 

J74 

87 

0.9986 

08 

4 

0.0698 

I75 

86 

0.9976 

IO 

5 

0.0872 

J74 

85 

0.9962 

14 

6 

0.1045 

J73 

84 

0.9945 

J7 

7 

0.1219 

X74 

83 

0.9925 

20 

8 

0.1392 

J73 

82 

0.9903 

22 

9 

0.1564 

172 

81 

0.9877 

26 

10 

0.1736 

172 

80 

0.9848 

29 

1  1 

0.1908 

172 

79 

0.9816 

32 

12 

0.2079 

171 

78 

0.9781 

35 

J3 

0.2250 

171 

77 

0.9744 

37 

14 

0.2419 

169 

76 

0.9703 

41 

15 

0.2588 

169 

75 

0.9659 

44 

16 

0.2756 

168 

74 

0.9613 

46 

*7 

0.2924 

168 

73 

o-9563 

50 

18 

0.3090 

1  66 

72 

0.9511 

52 

J9 

0.3256 

166 

71 

0-9455 

56 

20 

0.3420 

164 

70 

0-9397 

58 

21 

0-3584 

164 

69 

0.9336 

61 

22 

0.3746 

162 

68 

0.9272 

64 

23 

0.3907 

161 

67 

0.9205 

67 

24 

0.4067 

1  60 

66 

0-9*35 

70 

25 

0.4226 

J59 

65 

0.9063 

72 

26 

0.4384 

158 

64 

0.8988 

75 

27 

0.4540 

156 

63 

0.8910 

78 

28 

0.4695 

J55 

62 

0.8829 

81 

29 

0.4848 

J53 

61 

0.8746 

83 

3<> 

0.5000 

152 

60 

0.8660 

86 

31 

0.5150 

I5° 

59 

0.8572 

88 

32 

0.5299 

149 

58 

0.8480 

92 

33 

0.5446 

J47 

57 

0.8387 

93 

34 

0.5592 

146 

56 

0.8290 

97 

35 

0-5736 

144 

55 

0.8192 

98 

36 

0.5878 

142 

54 

0.8090 

IO2 

37 

0.6018 

140 

53 

0.7986 

104 

38 

0.6157 

139 

52 

0.7880 

106 

39 

0.6293 

136 

5' 

0.7771 

109 

40 

0.6428 

i35 

50 

0.7660 

in 

4i 

9.6561 

133 

49 

0-7547 

Ir3 

42 

0.6691 

130 

48 

Q-7431 

116 

43 

0.6820 

129 

47 

o-73M 

117 

44 

0.6947 

127 

46 

o-7J93 

121 

45 

0.7071 

124 

45°  * 

0.7071 

122 

Cosine 

£>i° 

1 

Sine 

D  i° 

TABLES. 

TABLE   L. 
International  Atomic  Weights  (Oxygen  =16), 


351 


Aluminum  
Antimony    

..      27.1 

.   .      I2O.2 

Magnesium    .... 
Manganese  

24.36 

55- 

Arsenic  

••      75- 

Mercury   

....    200. 

Barium  

•  •    J37-4 

Nickel  

....       58.7 

Bismuth  

.  .    208. 

Nitrogen  

....       14  01 

Boron  

ii. 

Oxygen  . 

16. 

Bromine  

.  .      79.96 

Phosphorus  

....      31. 

Cadmium  

.  .    1  12.4 

Platinum  

194.8 

Caesium  

•  •    i32-9 

Potassium    

....      39.15 

Calcium  -.  .  .  . 

.  .      40.0 

Radium  

226. 

Carbon   

.  .         12.  OO 

Selenium    

....      79.2 

Chlorine   

•  •      35-45 

Silver  

....    107.93 

Chromium   

52.1 

Silicon    

....      28.4 

Cobalt  

•  •      59- 

Sodium  

....      23.05 

Copper   

..      63.6 

Strontium    

....      87.6 

Fluorine   .  .  . 

.  .      19 

Sulphur    

....      32.06 

Gold   

.  .    197.2 

Tantalum  

....    181. 

Helium  . 

4 

Thorium  

....    232.5 

Hydrogen  .  . 

1.008 

Tin  

....    119. 

Iodine  '  

.       126.97 

Tungsten   

....    184. 

Iron    

55-9 

Uranium    

....    238.5 

Lead  

.'  .    206.9 

Zinc  

....      65.4 

Lithium   

7.03 

Reduction  of  Weighings  to  Vacuum.  (IV)  p.   23, 
Barometer  Correction.   (V)  p.  28. 


352 


TABLES. 


TABLE  LI. 

Density  and  Volume  of  One  Grain  of  Water  at  Different 
Temperatures. 


Temp. 

Density 

Vol.  of  i.  gr. 

Temp. 

Density 

Vol.  of  i.  gr. 

0° 

0.999878 

.000122 

21° 

0.998065 

.001939 

i 

°-999933 

.000067 

22 

0.997849 

.0021  56 

2 

0.999972 

.000028 

23 

0.997623 

.002383 

3 

Q-999993 

.000007 

24 

0.997386 

.00262  T 

4 

I.OOOOOO 

.000000 

25 

0.997140 

.002868 

5 

0.999992 

.000008 

30 

0-99577 

.00425 

6 

0.999969 

.00003  J 

35 

0.99417 

.00586 

7 

0.999933 

.000067 

40 

0.99236 

.00770 

8 

0.999882 

.0001  1  8 

45 

0-99035 

.00974 

9 

0.999819 

.000181 

50 

0.98817 

.01197 

10 

0-999739 

.000261 

55 

0.98584 

.01436 

1  1 

o  999650 

.000350 

60 

0-98334 

.01694 

12 

0.999544 

.000456 

65 

0.98071 

.01967 

13 

0-99943° 

.000570 

70 

0.97789 

.02261 

M 

0.999297 

.000703 

75 

0-97493 

.02570 

15 

0.999154 

.000847 

80 

0.97190 

.02891 

16 

0.999004 

.000997 

85 

0.96876 

.03225 

X7 

0.998839 

.001162 

90 

0.96549 

•03574 

18 

0.998663 

.001339 

95 

0.96208 

.03941 

19 

0.998475 

.001  527 

100 

0.95856 

•04323 

20 

0.998272 

.001731 

TABLE  LII. 

Density  of  Gases  (o°,  76  cm.).1 

'Hydrogen 00008987 

Oxygen 0014290 

Nitrogen 0012507 

Air    0012928 

Chlorine 003167 

Carbon  monoxide 0012504 

Carbon  dioxide 0019768 

Ethane 001341 

Ethylene 001252 

Steam  (at  100°) 00060315 

Largely  from  Guye,  J.  Ch.  Phys.,  1907,  p.  203. 


TABLES. 


353 


TABLE  LIII. 

Density  (o°),  Specific  Heat  (o°),  and  Coefficient  of  Linear  Expansion. 


Element 

»™*y  \  sifmc 

Coef.  of  Lin.  Exp. 
Multiplied  by  io6 

Aluminum   

2.60 

.22 

23.1 

Antimony     

6.62 

.049 

Bismuth  

9.8 

.031 

Cadmium   

8.61 

•055 

3°-7 

Carbon,  diamond  

3-52 

.IO 

1.18 

Carbon,   graphite  2.25 

•J5 

7.8 

Carbon,  gas  carbon  ...          1.90 

5-4 

Cobalt     8.8 

.106 

12.4 

Copper    

8.92 

.094 

16.8 

Gold   

19-3 

.032 

14.4 

Iron    

7.8 

.1  1 

12.  I 

Lead  

1  1.36 

.029 

29.2 

Magnesium 

i  74 

Mercury   

I3-596               -0333 

181    (cub. 

exp.) 

Nickel  

8.9 

.108 

12.8 

Phosphorus,  yellow.  .  . 

V 

1.83 

.20 

Phosphorus,   red  

2.19 

-1? 

Phosphorus,  metallic  . 

2    T.A. 

Platinum  

*  •O'T 
21.4 

•°33 

9.0 

Silver  

IO.  S3 

.0  =;6 

19.2 

Tin  

JO 

7-3 

v  o 
.056 

22.3 

Zinc    7.2 

.094 

29.2 

354 


TABLES. 
TABLE  LIV. 


Density,  Specific  Heat,  and  Coefficient  of  Cubical  Expansion  of  Liquids. 
(o°  unless  otherwise  specified.) 


Liquid 

r^       .,             Specific        Coef.  of  Cub.  Exp. 
Density            J^            Multiplied  by  ii 

Acetone 

.792 
1.080 

.8! 

•83 
.81 

•956 
1.04 
.899 
1.293 
i-53 
•74 
•905 

1.2(4 

.64 

.914 
464 

•07 
•9 
i-i35 
•79 
1.025 

•  J 

•52 
•54 
.69 

•59 

"I" 

•23 
•53 

".58" 

•348 

•057 
.048 

.89 

•85 
.28 
.817 
.176 
.14 
.1  1 

•51 

.28 

Acetic  acid  
Ethyl  alcohol 

Amyl  alcohol  
Methyl  alcohol  
Methyl  acetate  
Aniline 

Benzol 

Carbon  bisulphide  .... 
Chloroform  

Ethyl  ether  
Ethyl  acetate  (17°).  .? 
Glycerine 

Ammonia  (liq  ) 

Carbon  dioxide  (o°)  .  .  . 
Carbon  dioxide  (31.3°). 
Hydrogen  (liq.  ,  —  252°) 
Oxygen  (liq.,  —  130°)  .  . 
Oxygen  (liq.,  -183°)  .  . 
Nitrogen  (liq.,  —  145°)  . 
Air  (-liq.,  —  190°)  

6.                            24. 
4-6 

•43                           5-6 

TABLE  LV. 

Density,    Specific    Heat,    and   Coefficient    of    Expansion    of 
Miscellaneous  Substances  (o°). 


Substance 

Density 

Specific 
Heat 

Coef.  of  Lin.  Exp. 
(Xio*) 

Castor  oil  

969 

Cork   . 

Glass,  green 

2    6 

I  O 

8  o 

Glass,  crown 

2    7 

I  O 

8  8 

Glass,  crystal  .  . 

2   o 

18 

7   7 

Glass,  flint  
Hard  rubber  

3-15~3-9 
i.i  < 

.19 

7-3 
7-7 

Marble  .  . 

2    7  C 

T    I      7 

Paraffin    

J-75 
.89 

Quartz,  crystal  II  .... 
Quartz,  crystal  J_   .  .  .  . 
Quartz,  fused  .... 
Wax   

2-653 
2-653 

2.20 
0  7 

.19 

7.2 
I3.2 

•54 

TABLES. 


355 


.11 
ii 


CC  00    O      .    t^ 
CM    ON  rj-     •    t^ 


0  O    O  •    O 

O  <N     O  •     M 

M  co  t^  •  00 

10  M       M  • 


r^    .    i-i      .   i>»          .Q         -3-00    co    • 

CO      •    M      .    ON  -00        CO    t^OO      •      •        00 


vO      -00      -00 
M     •    r#5    •   H 


M     1-1 
0)     CO 


^  O    «•  t^  «    YJ       00 
T  10  co  CO  VO  ^  t^ 


O    •*  O 

O  00  O 

O  CO  t^. 

00    rj-  10 


.    10 

.   co 


to 


P  S 


1«l 

<U  +J  Q. 

Q  rt 


bjO 

.5 

N 


ng  P 


00  •  CM 

0  OO    vo  cs    t^.oO          vo\O    M    H    M  OO          Tj-MOCOOOt^-OO 

N     M     M     Hi     H     N              MMMM  COCOWNNWMN 

1  I        I        I        I        I 

I>-COON!IH.         NO.      ..I  |      ]             '.'.'.   ."'.*'.'. 

O    H    t^     .    «       .  <^-....  .      »  

VO    COOO 

O  W  OO   '           Cl  t**»  VO  CO  "  t^»  VO  M  CO  VO  *•/}  O  ON 

0  vo  *3-  '          00  O  ON  ON  "  COOO  O  ON  W  00  ON  VO 

N                                                                                          N  VO               (S                 M 

vovo                 oo             \OVOVON  cow't             M 

01  H  T!-  i-t  o  oo        d  cooo  <soo  co'        000000^0^0000 

VOOO    ON  ONOO  ^O          n    co  co  CM    r^  O                 O    H    t^OO    VO  T^  co^O 

rrrrrr  |M   '  r : 

O            00--  ••          OcoO 

.      .           .      .    o      •      •  •      •        00    Tj-         co 

»       m     _         ••!•(•• 

oSSloo-        -^        ••  ••                   r*' 

^vSo^  >{S   '      '21C''  ''     o^1^ 

I  *>.vo'  ;  I    ;       ;  I   I    ;   ;  ;   ; 

i  i    :      :      :.       :  :  :  : 

•       '     w'   •   *'x  «  '   '       '.       '   '.   !iS 

'x    !    !  «-<'p  !    ! 

Ill  lag    ll||§  -11 


356 


TABLES. 


Gases,  Vapors,  and  Liquids. 

Deviations  from  Boyle's  Law  (XI),  p.  97 

Coefficient  of  Increase  of  Volume  of  Gas  (XII),  p.  98. 

Coefficient  of  Increase  of  Pressure  of  Gas  (XIII),  p.  88. 

TABLE  LVII. 

Pressure  of  Saturated  Water  Vapor  (Regnault). 

(mm.) 


Temp. 

Pressure 

Temp. 

Pressure 

Ice                   Water                   29° 

29.782 

30 

31-548 

—  10 

1.999                   2.078                   31 

33-405 

8 

32 
2-379                  2.456                   33 

3  5  3  59 
37-4io 

34 

39  565 

6 

2.821                   2.890                   35 

41.827 

40 

54.906 

4 

3-334-                  3-387                    45 

r  —  . 

7I-39I 

5° 

Q  I  .  Q  o  2 

2 

3-925                  3-955                   55 

117.479 

1          60 

148.791 

65 

186.945 

O 

4.600 

70 

233-093 

+      I 

4.940 

75 

288.517 

2 

5-302 

80 

354-643 

3 

5-687 

85 

433-41 

4 

6.097 

90 

525-45 

5 

6-534 

545-78 

6 

6.998 

92 

566.76 

7 

7.492 

93 

588.41 

8 

8.017 

94 

610.74 

9 

8-574 

95 

633-78 

10 

9.165 

96 

657-54 

1  1 

9-792 

97 

682.03 

12 

10.457 

98 

707.26 

13 

1  1.062 

98-5 

720.15 

14 

11.906                            99-o                 733-91 

15 

12.699                            99-5 

746.50 

16 

13.635                           100.  o 

760.00 

1  7 

14.421                    100.5 

773-71 

18 

J5-357                          IOI-° 

787-63 

19 

16.346                                       102.0 

816.17 

20 

17.391                                       104.0 

875-69 

21 

18.495 

105 

906.41 

22 

19.659 

I  10 

1075-4 

23 

20.888 

I  20 

149  1.3 

24 

22.184 

I30 

2030.3 

25 

23-55°                         I5° 

3581.2 

26 

24.998                         175 

6717 

27 

26.505                                      200 

1  1690 

28 

28.101                                       225 

19097 

TABLES. 

TABLE  LVIII. 
Vapor  Pressure  of  Mercury  (mm.). 


357 


JL  Clllp. 

J.  V-11J.L/. 

o 

O.O2 

170 

8.091 

+  20 

O.O4 

i  So 

I  l.OOO 

40 

0.08  ' 

190 

14.84 

60 

o.'i6 

200 

19.90 

80 

o-35 

210 

26.35 

IOO 

0.746 

220 

34  7° 

I  IO 

I-°73 

230 

45-35 

120 

i-534 

240 

58.82 

130 

2-i75 

250 

75-75 

140 

3.059                         260 

96.73 

J5° 

4.266                         270 

123.01 

1  60 

5.900                         280 

I55-I7 

358 


TABLES. 

TABLE    LIX. 
Vapor  Pressure  of  Alcohols. 

(Pressure  in  cm.  of  Mercury.) 


Temperature 

Ethyl  * 

Propyl  2 

Isobutyl? 

Amyla 

—  10 

6  1 

o 

1.2  < 

.-1  Z 

10 

2.42 

.78 

•4 

20 

4-47 

1.48 

•9 

qo 

7  n  •} 

2   8l 

I    7 

40 

13-53 

5-n 

3-2 

•9 

5° 

22.26 

8.90 

5.6 

1.8 

60 

35-38 

14.90 

9.6 

3-2 

70 

54.67 

24.04 

15.8 

5.6 

80 

81.71 

37-53 

25.1 

9-4 

90 

1  19.64 

56.81 

38-7 

i  5.2 

IOO 

!7°-33 

83-58 

57-8 

23-7 

I  I  O 

237    3  <\ 

11982 

•3  c  7 



120 

323-47 

167.70 

52.2 

130 

433-1 

229.6 

74-4 

140 

C7i.i 

308 

IOV3 

Batslli.     2  Ramsey  and  Young.     3  Schmidt. 


TABLES. 

TABLE  LX. 
Vapor  Pressure  of  Benzol  (Benzene),  Regnault. 


359 


Temp. 

Pressure  (cm.) 

Temp. 

Pressure  (cm.) 

—  20 

—  10 

•58 

1.29 

70 
80 

54-74 
75-19 

0 

2-53 

90 

101.27 

IO 

4-52 

IOO 

134.00 

20 

3° 
40 

12.  02 
18.36 

I  IO 
120 
I30 

174.4 

223-5 
282.4 

5°       . 
60 

27.14 
39-01 

I4O 

352 
433 

TABLE   LXI. 
Vapor  Pressure  of  Carbon  Bisulphide  (Regnault). 


Temp. 


Pressure  (cm.) 


Temp. 


Pressure  (cm.) 


—  20 

4-73 

50 

85-7I 

—  10 

7-94 

60 

116.45 

0 

12.79 

7° 

!55-2I 

10 

19.85 

80 

203.25 

20 

29.80 

90 

261.91 

3° 

43-46 

IOO 

332.  51 

40 

6i-75 

J5° 

909.6 

* 

Comparison  of  Corresponding  States  (XIV),  p.  108. 
Surface  Tension  of  Liquids  (XV),  p.  117. 
Coefficient  of  Viscosity  (XVI),  p.  120. 


Solutions. 

Coefficient  of  Absorption  (Henry's  Law)  (XIX),  p.  151. 

Solubility  of  Partially  Miscible  Liquids  (XX),  p.  154. 

Valson's  Moduli  (XXI),  p.  159. 

Density  of  Ammonium  Chloride  (XXII),  p.  160. 

Coefficient  of  Diffusion  (XXIII),  p.  168. 

Molecular  Elevation  (XXIV),  p.  174. 

Molecular  Lowering  (XXV),  p.  177 


Thermochemistry. 


Specific  Heat  of  Water  (II),  p.  5. 

Specific  Heat  of  Gases  (XVII),  p.  126. 

Molecular  Specific  Heats  at  Different  Temperatures  (XVIII),  p.  1 27. 


360  TABLES. 

TABLE  LXII. 

Melting  Point  of  Metals.     (Holborn  and  Day  and 
Waidner  and  Burgess.1) 

Tin 232 

Cadmium 321 

Lead 327 

Zinc 419 

Antimony ". 631 

Aluminum 657 

Silver 961 

Gold 1063 

Copper 1084 

Platinum 1770 

Heat  of  Formation  (XXVI),  p.  184,  194. 
Heat  of  Dilution  (XXVII),  p.  186. 
Heat  of  lonization  (XXVIII),  p.  192. 
Heat  of  Combustion  (XXIX),  p.  194. 

Light. 

Wave  Lengths  of  Different  Types  of  Ether  Waves  (XXX),  p.  208. 
iPhys.  Rev.,  1909,  p.  467.     C.  R.,  1909,  cxlviii,  p.  1177. 


TABLES. 

TABLE  LXIII. 
Wave  Lengths  in  Angstrom  Units  (io-8  cm.). 


36i 


Line 

Element 

Wave  Length 

Color 

c,  Ha  -;\J; 

Dz 

Hydrogen 
Sodium 

6563.054 
c8o6  i  ^  ? 

Red 
Yellow 

D2  ! 
F.H0  
G'  Hr.. 

Sodium 
Hydrogen 
Hydrogen 

5890.182 
4861.527 
4.  74.0  634. 

Yellow 
Blue 

Violet 

T 
H  

Calcium 
Helium 

3968.625 
7O6  Z..2 

Violet 
Red 

Helium 
Helium 

6678.1 

cg7  c  6 

Red 

Yellow 

Helium 
Helium 

50!5-7 

AQ2  I    Q 

Green 
Blue 

Helium 
Helium    * 

47I3'2 
A.A.7  I    c; 

Blue 
Violet 

Mercury 

Red 

Mercury 
Mercury 

5790-7 

Yellow 
Yellow 

Mercury 
Mercury 

31  v*t-v 

5460.7 

Green 
Green—  Blue 

Mercury 
Mercury 

4y  iy-  / 
4916.4 

AT.  £8    T. 

Blue 
Blue 

Mercury 

<+o  3°-6 
407  8  i 

Violet 

Mercury 

Violet 

K,v 

Potassium 

7  6OO   3 

Red 

KB. 

Potassium 

<:8^2.2 

Yellow 

Kr. 

Potassium 

4.O4.  7  4. 

Violet 

T    ^ 

Li- 

Lithium 

6708  2 

Red 

Li/?  . 

Lithium 

6107  8 

Orange 

Cadmium 

6/178  c 

Red 

Cadmium 

U4o°-  o 
508  s-  8 

Green 

Cadmium 

Blue 

1 

Light  Filters  (VIII),  p.   146. 
Refractive  Indices  (XXXI),  p.   214. 
Atomic  Refractivities  (XXXII),  p.  215. 
Specific  Rotatory  Power  (XXXIII),  p.   219. 
Magneto-optic  Rotation  (XXXIV),  p.  220. 


362  TABLES. 

Electro-chemistry. 

Conversion  of  Practical  to  C.  G.  S.  Units  (III),  p.  8. 

Slide  Wire  Bridge  (Ratios)   (IX),  p.   51. 

Specific  Conductivity  of  Potassium  Chloride  Solutions  (X),  p.  57 

Temperature  Coefficient  (XXXVII),  p.   290. 

Equivalent  Conductivities  (XXXV),  p.   288. 

Transport  Numbers  (XXXVIII),  p.   294. 

Ionic  Conductivities  (XXXIX),  p.  295. 

Degree  of  Dissociation  (XXXVI),  p.  289. 

Dissociation  Constant  (XL),  p.  303. 

Dissociation  of  Water  (XLV),  p.  327. 

Conductivity  of  Non-aqueous  Solutions  (XLI),  p.  308. 

Conductivity  of  Fused  Silver  Nitrate  (XLII),  p.  309. 

Dielectric  .Constants  (XLIII),  p.  311. 

Electrode  Potentials  (XLIV),  p.  318. 

Production  of  Ozone  (XLVI),  p.  340 

Radioactive  Transformations  (XLVII),  p.  343 


INDEX. 


Abbe,  refractometer,  212,  216 
Abbreviated  arithmetic,  14 
Absolute  temperature,  86,  87,  141 

zero,  86,  142 
Absorbing  power,  198 
Absorption  spectra,  207-209 
Accumulators,  336,  337 
Aceto-chloranilide,  228 
Acids,  strength  of,  229,  231 
Active  mass,  226 

substances,  optically,  217 
Additive  properties,  158 
Adiabatic  changes,  127,  129,  130 
Affinity  constant,  256,  303 

of  acid    229 

of  base,  237 

Alcohols,  vapor  pressures,  358 
Alloys,  271,  272,  276 
Amagat,  gas  laws,  97,  98 
Amalgams,  309,  322 
Ames,  mechanical  equivalent,  284 
Ammonia,  solutions  in  liquid,  308 
Ampere,  definition,  7 
Angstrom  unit,  3 
Anion,  definition,  279 
Anode,  definition,  279 
Approximation  formulae,  15 
Arrhenius,  conductivity  cell,  55 

degree  of  dissociation,  289 

dissociation,  281 

vapor  pressure,  170 
Association,  162,  178 
Atomic  dispersion,  216 

heats,  124 

refractivities,  215 

weights,  351 
Avogadro's  law,  89 
Aulich,  dilution  law,  261 

Balance,  air  buoyancy,  22 

correction  of  weights,  23 

directions  for  use,  20 

ratio  of  arms,  22 

weighing  by  oscillations,  21 
Barometers,  27 

corrections,  28 


Bases,  strengths  of,  237 
Batschinski,  mean  density,  113 
Beckmann,  boiling  point  apparatus,  35 

freezing  point  apparatus,  35 

thermometers,  33 

Bender,  von,  refractive  indices,  160 
Berkeley  and  Hartley,  osmotic  pres- 
sure, 163 
Berthelot,  calorimeter,  38 

principle  of  maximum  work,  186 
Billitzer,  null  electrodes,  315 
Bimolecular  reactions,  235-240 
Black  body,  198,  199 
Boiling,  in 
Boiling  point,  apparatus,  Beckmann, 

35,  175 

elevation  of,  173 

Landesberger- Walker,  36,  175 

of  liquids,  355 

of  liquid  mixtures,  153-175 

of  water,  33 
Bolometer,  34 
Bomb  colorimeter,  39     ' 
Boyle's  law,  79,  82 

deviations,  96 
British  thermal  unit,  4 
Briihl,  atomic  refractions,  216 
Buckingham,    absolute  temperatures, 

142 

Bunsen,  effusion  apparatus,  95 
Burette,  calibration,  76 
Cadmium  cell,  335 

Calcium  carbonate,  dissociation,  257, 
.    268,  272 

chloride,  heat  of  formation,   189 
Callender  and  Griffith's  bridge,  53 
Calomel  electrode,  60 
Calorie,  definition,  4 
Calorific  power,  5,  193-197 
Calorimeter,  Berthelot's,  38 

bomb  (Hempel's),  39 

gas  (Junker),  42,  196 

liquid  (Junker),  43 

fuel  (Rosenhain),  40 

water  equivalent,  39,  40,  41 
Cane-sugar,  inversion,  231,  232 


363 


364 


INDEX. 


Cane-sugar,  osmotic  pressure,  163 
Capacity,  definition,  6 
Capillary  electrometer,  59,  319,  320 
Carbon-dioxide-free  vessels,  77 
Carbon  monoxide,  heat  of  formation, 

189 

Carnot  cycle,  138-140 
Catalysis,  229,  231 

of  ester,  229 
Cathetometer,  19 
Cathode,  definition,  279 
Cation,  definition,  279 
Cell,  constant  of,  57 

energy  of,  320,  331 

primary,  334,  336 
Charge  on  ions,  281 
Charles'  law,  85 
Chemical  dynamics,  225-249 

equilibrium,  246-254 

equivalent,  280 

statics,  250-278 
Clapeyron's  equation,  146 
Clark  cell,  335 
Clausius,  equation,  147 

kinetic  theory,  87 

thermodynamics,  137,  143 
Claude,  liquid  air,  131 
Cleaning  glass,  78 
Clement    and    Desormes'    apparatus, 

i33-*36 
Coefficient  of  absorption,  150-152 

cubical  expansion,  354 

diffusion,  168 

increase  of  pressure,  98 

increase  of  volume,  98 

linear  expansion,  353,  354 
Coefficient,  temperature  of  cell,  329 

of  conductivity,  290,  306 
Comparator,  18 
Compensation  method,  57 
Component,  phase  rule,  268 
Concentration  cells,  321-324 
Conductivity,  cells,  55 

equivalent,  288,  308  . 

maximum,  307 

molecular,  286 

of  fused  salts,  309 

of  non-aqueous  solutions,  307 

specific,  8 

and  temperature,  306 
Connectors,  clip,  78 
Conrody,  atomic  refractions,  216 
Constantin,  wire,  34 
Conversion  factors,  electrical  units,  8 
Copper  voltameter,  64-66,  283 
Corpuscle,  339,  341,  342 


Corresponding  states,  106,  108 
Cosines,  350 
Coulomb,  definition,  7 
Coulometer,  64,  282 
Critical  angle,  210 

data,  355 

pressure,  104 

state,  103,  1 08 

temperature,  104 

volume,  104,  113 
Cryohydrates,  275 
Cubical  expansion,  354 

Dal  ton's  law,  84,  102,  153 
Daniell  cell,  335 

Degree  of  dissociation,  287,  289,  308 
Density,  definition,  3 

of  gases,  352 

of  liquids,  24,  354 

of  solids,  353,  354 
De  Vries,  osmotic  pressure,  163 
Dew  point,  29,  103 
Diazonium  salt,  decomposition,  234 
Dickinson  and  Mueller,   sodium  sul- 
phate, 277 

Dielectric  constant,    and   dissociation, 
310 

definition,  6 

measurement,  66,  311,  312 
Difference  of  potential  of   cells,  329, 

334 

of  electrodes,  313,  334 
Diffraction  grating,  48 
Diffusion,  167 
Dilatometer,  277 
Dilution  law  of  Kohlrausch,  303,  306 

of  Ostwald,  256,  302 

of  Rudolphi,  302 
Dispersion,  refractive,  216 

rotatory,  220 
Displacement  law,  201 
Dissociation,  162,  178 

constant,  256,  302,  303 

degree  of,  287,  289,  308 

electrolytic,  281,  285 

of  gases,  257,  265-267 
Distillation  of  liquid  mixtures,  154,  157 
Distribution  constant,  261 

law,  260    . 

Dolezalek  electrometer,  63 
Donnan,  order  of  reaction,  245 

osmotic  pressure,  164 
Dropping  electrode,  315,  317 
Drude,  dielectric  constant,  68 
Dry^cell,  336 
Drying  tubes,  74 


INDEX. 


Dulong  and  Petit's  law,  124 
Dumas,  vapor  density,  91 
Dyne,  definition,  4 

Edison  cell,  336 

storage  cell,  337 
Efficiency,  141 
Effusion,  94,  95 
Electric  charge,  6,  7,  64 

charge  on  ions,  281 

current,  6,  63 

furnace,  203 

quantity,  64 
Electrical  measurements,  49 

thermometers,  34 

units,  5,  8 

Electro-capillary  phenomena,  317 
Electro-chemical  equivalent,  279,  280 
Electrode,  calomel,  60 

classes,  319 

dropping,  315 

gas,  328 

null,  315 

potentials,  313-334 
Electrolysis,  285 
Electrolytes,  equilibrium,  254 

resistance,  54 

Electrolytic  conductivity  at  high  tem- 
perature, 306 

conductivity  of  solids,  310 
Electromagnetic  units,  5 
Electrometer,  capillary,  59,  319,  320 

Dolezalek,  63 

electrostatic,  63 
Electromotive  force,  measurement  of, 

57 

small  known,  63 

unit  of,  6,  7 

Electrons,  339,  341,  342 
Electrostatic  units,  6 

voltmeter,  63 

Emanation  from  radium,  342 
Emissive  power,  198 
Emptying  bulb,  77 
Endosmosis,  170 
Endothermic  reaction,  187 
Entropy,  142 
Eotvos'  law,  115 
Equilibrium,  constant,  246 

effect  of  temperature  on,  250,  263 

of  electrolytes,  254 

principle  of  stable,  183 
Equivalent  conductivity,  8,   286,   288, 
291,  308 

conductivity   at   infinite   dilution, 
301 


Erg,  definition,  4 

Errors,  estimation  of,  10-14 

Ester,  catalysis  of,  229 

saponification  of,  237-240,  299 
Eutectic  alloy,  272,  276 
Ewell,  ozone,  340,  345 

rotatory  polarization,  217 
Excited  activity,  343 
Exner,  molecular  refractivity,  215 
Exothermic  reaction,  187 
Expansion  of  liquids,  354 

of  solids,  353,  354 
Exponential  equations,  9 

formula,  228 

Earad,  definition,  8 

Faraday,  law  of,  279 

Fery  pyrometer,  202 

Frick's  law  of  diffusion,  167 

Filling  bulb,  77 

First  law  of  thermodynamics,  123 

order  reactions,  227-233 
Fischer,  F.,  ozone,  340 
Fluidity,  of  fused  salts,  309 
Foote,  regulator,  72 
Fractional  distillation,  157 
Fredenhagen,  spectra,  206,  207 
Free  energy  equation,  144 
Freezing  point,  apparatus,  35 

effect  of  pressure,  145 

lowering,  176 

of  liquids,  355 

molecular  weights,  176-179 
Fused  salts,  conductivity,  309 

Gas,  burette,  72 

condensing  vessel,  73 

electrodes,  328 

equation,  86 

seal,  73 
Gases,  density,  352 

dissociation,  257,  265-267 

electrical  conductivity  of,  339- 
346 

kinetic  theory  of,  80-82 

liquefaction  of,  103,  131-133 

solubility  in  liquids,  150-152 

specific  heats,  124-127 

spectra,  206,  209 
Gay  Lussac,  law  of,  85,  88 

deviations,  97-99 
Gibbs,  phase  rule,  267-275 

potentials,  225 

theory  of  cell,  331 

Goodwin,  conductivity  of  fused  salts, 
3°9 


366 


INDEX. 


Gram  atom,  3 

equivalent,  3 

molecule,  3 

Gray,  absorption  vessels,  345 
Groshans,  molecular  volume,  160 
Guldberg  and  Waage,  mass  law,  247, 

.253 
Guthe,  silver  voltameter,  282 

Hagenbach,  sulphur  dioxide  solutions, 

3°7 
Heat  of  combustion,  5,  193,  197 

of  dilution,  183,  186 

of  dissociation,  265 

of  formation,  181,  184-186,   191, 
.  320  t 

of  lomzation,  192 

of  neutralization,  191,  193 

of   solution,    182,    184-188,    264, 
265 

of  vaporization,  191,  263 

units,  4 
Helium,  liquefaction,  133 

from  radium,  341 
Helmholtz,  double  layer,  316 

dropping  electrode,  315 

energetics  of  cell,  331 
Hempel,  calorimeter,  39 

gas  burette,  72 
Henry's  law,  150 
Hess,  law  of,  188 

thermo-neutrality,  190 
Heydweiller,  dissociation  of  water,  327 
Hittorf,  velocity  of  ions,  291 
Holborn-Kurlbaum  pyrometer,  202 
Horsepower,  4 
Hydrates,  306 
Hydrogen  ions,  catalytic  action  of,  229, 

231 
Hydrolysis,  general,  257 

weak  base,  232 

weak  acid,  239 
Hygrometer,  chemical,  31 

Regnault,  29 

wet  and  dry  bulb,  29 
Hygrometry,  29,  103 

Ice,  vapor  pressure,  356 
Incomplete  reactions,  246-249 
Indicators,  theory,  256 
Internal  energy,  123,  145,  182 
Invariant  system,  271,  272 
Inversion  of  cane-sugar,  231,  232 
Ionic  conductivity,  295 
lonization  constant,  255,  303 
heat  of,  192 


Ions,  charge  on,  281 

gaseous,  339 

velocity  of,  286,  291,  294,  296,  298 
Irreversible  processes,  142 
Isochore  equation,  Van't  Ho  IT,  253 
Isohydric  solutions,  304 
Isothermal  cycle,  140 
Isothermals,  99,  108 
Isotonic  solutions,  164 

Jager  and  Steinwehr,  cadmium  cell,  335 

water  equivalent,  39 
Jahn,  dilution  laws,  302 

absorption  spectra,  208 
Jones,  hydrates,  306 

mixed  solvents,  309 

non-aqueous  solutions,  308 
Joule,  definition,  4 
Joule's  law,  283 
Junker  calorimeter,  42,  196 

Kahlenberg,  Faraday's  law,  280 

non-aqueous  solutions,  307 

theory  of  solution,  183 
Kammerlingh-Onnes,  Boyle's  law,  96 

liquefaction  of  helium,  133 
Kelvin,   absolute   temperature     scale, 
141 

dropping  electrode,  315 

thermodynamics,  137 
Kilowatt,  definition,  4 
Kinetic  energy  of  molecules,  87 

theory  of  gases,  80-82 
Kirchoff,  radiation  law,  198,  207 
Kohlrausch,    degree    of    dissociation, 
289 

dilution  law,  303 

ionic  conductivity,  295 

temperature  coefficient,  290,  306 
Konowalow,  liquid  mixtures,  154 
Kraus,  alcohol  solutions,  307 

ammonia  solutions,  307,  308 
Kundt,  velocity  of  sound,  128 
Kurlbaum,  pyrometer,  202 

radiation  constants,  199 
Fused  salts,  309 

Landesberger-Walker,    boiling    point 

apparatus,  36,  175 
Landolt,  light  filters,  46 
Latent  heat,  effusion,  355 

of  vaporization,  no,  114,  355 
Lebedew,  radiation  pressure,  200 
Le  Blanc,  polarization,  332 
Le  Chatelier,  calcium  carbonate,  257 

law  of  equilibrium,  187 


INDEX. 


367 


Le  Chatelier,  pyrometer,  202 

specific  heats,  127 
Lerlanche  cell,  336 
Lewis,  mass  and  velocity,  342 

thermodynamics,  123 
Light,  198-224 

emission  of,  198-202 

filters,  46 

monochromatic,  45 

plane-polarized,  217 

velocity  of,  210 

wave  lengths,  361 
Linde,  liquefaction  of  gases,  132 
Lippich,  polarimeter,  222 
Lippmann,  capillary  electrometer,  319 
Liquefaction  of  gases,  103,  131-133 
Liquid  cells,  326 
Lodge,  velocity  of  ions,  294 
Logarithms,  common  and  natural,  9 

table,  348 

transformations,  10 
Loomis,  degree  of  dissociation,  289 
Lorentz-Lorenz,  refractivity,  160 
Lubricant,  cock,  78 
Lummer,  radiation,  200,  201 
Luminescence,  chemical,  198,  206 

electric,  198,  206 

Magnetic  force,  unit  of,  5 

rotation,  219 
Manometers,  26,  27 
Mass  action,  law  of,  225-227,  247,  253 
Mass  and  velocity,  342 
Masson,  velocity  of  ions,  294 
Mathias'  rule,  112 
Maximum  work,  principle  of,  186 
Maxwell,  radiation  pressure,  200 

viscosity,  119 

Measuring  vessels,  calibration,  75 
Mechanical  equivalent  of  heat,  1 23 
Mechanical  units,  4 
Melting  points  of  metals,  360 
Membranes,  semi-permeable,  163 
Mercury,  purification,  74    . 

still,  74 

Meyer,  Victor,  vapor  density,  93 
Micrometer  microscope,  18 
Micron,  definition,  3 
Mixed  solvents,  conductivity,  309 
Mohr-Westphal  balance,  25 
Molecular  conductivity,  286 

dispersion,  216 

elevation,  174 

gas  equation,  107 

heat,  124 

lowering,  177 


Molecular,  refractivily,  214 
rotatory  power,  219 
weights,  in  solution,  178 
of  liquids,  115,  118 
of  gases,  90,  95 
of  vapors,  91-94 

Morse,  osmotic  pressure,  163,  169 
Monomolecular  reactions,  227-233 

Natanson,  nitrogen  peroxide,  257 
Nernst,  dilution  laws,  302 

dielectric  constant,  68 

dissociation  of  water,  187 

distribution  law,  261 

free  energy,  145 

glower,  310 

hydrates,  306 

molecular  weights,  94 

solution  pressure,  313 
Nernst-Thomson  rule,  311 
Neutralization,  heat  of,  191,  193 
Nichols  and  Hull,  radiation  pressure, 

200 

Nitrogen  oxides,    absorption   spectra, 
207 

dissociation,  257-260 

electrical  production,  339-341 
Non-aqueous  solutions,  307 
Normal  electrode,  60 

solution,  3 
Noyes,  ethyl  alcohol,  155 

conductivity     arid     temperature, 
290,  306 

dilution  laws,  303 

maximum  conductivity,  307 
Null  electrode,  315 

Ohm,  definition,  7 
Olszewski,  Boyle's  law,  96 
Optically  active  bodies,  217 
Optical  pyrometers,  202-205 
Order  of  a  reaction,  240—243 

of  one  component,  243-246 
Osmotic  pressure,  experimental  results, 
163,  169 

thermodynamic  study,  165-167 
Ostwald,  catalysis  of  ester,  229 

degree  of  dissociation,  289 

dilution  law,  256,  302 

ionic  conductivity,  295 

order  of  reactions,  243 

pyknometer,  25 

Oxygen,  separation  from  nitrogen,  132 
Ozone,  electrical  production,  339,  340, 
344-346 

thermal  production,  266,  340 


368 


INDEX. 


Oudemann,  rotatory  polorization,  161, 
219 

Palmaer,  electrode  potentials,  319 
Partition  Jaw,  261 
Period  of  reaction,  228 
Pfeffer,  osmotic  pressure,  163 
Phase,  definition,  268 

rule,  267-269 

applications,  268-275 
Plane-polarized  light,  217 
Planck,  liquid  cell,  326 

radiation  law,  201 
Platinum-black,  56 

resistance  thermometer,  34,  205 
Polarimeters,  220-224 

bi quartz,  221 

half -shade,  222 
Polarization,  electrolytic,  332-334 

rotatory,  217-224 

Potassium     choride     solutions,     con- 
ductivity, 57 
Potential,  chemical,  225,  269 

of  electrodes,  313-334 

measurements,  57-62 

signs,  62 

Potentiometer,  57-59 
Prefixes,  meaning  of,  8 
Pressure,  measurement  of,  26 

of  light,  200 

units,  4 

Prism  spectrometer,  47 
Pulfrich,  refractometer,  211,  216 
Pyknometers,  24 

Radiation  correction,  43-45 

pressure,  200 

pyrometers,  202-205 

thermal,  198-202 
Radioactivity,  '341-343 
Radium,  341,  343 
Raoult,  vapor  pressures,  172 
Ratio  of  specific  heats  of  gases,   125, 

128 

Refraction,  index  of,  210 
Refractivity,  atomic,  215 

molecular,  214 
Refractometers,  211-216 
Regnault,  hygrometer,  29 

vapor  pressures,  in 
Relative  humidity,  103 
Resistance,  box,  49 

electrolytic,  54 

measurement,  50-57 

specific,  8 

unit  of,  6,  7 


Reversible  processes,  140 
Richards,  Faraday's  law,  280 

electrochemical    equivalent    of 

silver,  281 

Ritchie,  radiation,  198 
Roberts-Austen,  alloys, 
Roozeboom,  alloys,  272 
Rosenhain,  calorimeter,  40 

alloys,  276 
Rotation,  magnetic,  219 

specific,  218 
Rotatory  polarization,  217-224 

dispersion,  220 
Rudolphi,  dilution  law,  203 
Rutherford,  radioactivity,  342 

Saccharimeters,  223 

Saponification  of  esters,  237-240,  299 

Second  law  of  thermodynamics,  137 

Second  order  reactions,  235—240 

Semi-permeable  membranes,  163 

Share  of  transport,  293,  294 

Siemen's  ozonizer,  345 

Silent  discharge,  340 

Silver  voltameter,  282 

Sines,  table,  350 

Single  potential  differences,  60-62 

Skaupy,  amalgams,  309 

Slide  wire  bridge,  50 

Sodium  flame,  45 

Solids,  electrolytic  conductivity  of,  310 

Solid  solutions,  178 

Solubility  constant  or  product,  325 

measurement  of,  324. 
Solution,  heat  of,  182,  18^4-188,  264, 
265 

pressure,  313 
Solutions.  150-180 

additive  properties  of,  158 

colloidal,  178 

of  gases,  150-152 

of  liquids,  153-157 

of  solids,  158 

solid,  178 

Sound,  velocity  of,  128 
Specific  conductivity,  8,  291 

heat,  of  gases,  124-127 
of  liquids,  354 
of  solids,  124,  353 
of  water,  5 

inductive    capacity,  6,    66,    310- 
312 

resistance,  8 

rotation,  218 
Spectra,  absorption,  207-209 

of  gases,  206,  209 


INDEX. 


369 


Spectra,  of  vapors,  206 
Spectrometer,  adjustments,  46 

grating,  48,  209 

prism,  47 

Stable  equilibrium,  principle  of,  183 
Standard  conditions  for  gases,  3 
Steele,  velocity  of  ions,  295 
Stefan-Boltzman  law,  199 
Storage  cells,  336,  337 
Stirrers,  72 

Straight  diameter,  law  of,  113 
Strengths  of  acids,  229 

of  bases,  237 
Surface  tension,  114,  117 

measurement  of,  116,  117 
Symbols,  table,  i 
Temperature,  absolute,  86,  141 

Temperature  coefficient,  of  cell,  329 
of  conductivity,  290 

effect  of,  on  conductivity,  306 

on  equilibrium,  250,  263 

on  heat  evolution,  190 

units,  4 
Thermal  neutrality,  190 

radiation,  198-202 
Thermochemistry,  181-197 
Thermocouple,  34,  203 
Thermodynamics,  123-149 

first  law,  123 

second  law,  137 
Thermometers,  Beckmann,  33 

bolometer,  34 

calibration,  31 

mercury,  31 

platinum  resistance,  34 

thermocouple,  34 
Thermoneutrality,  190 
Thermostats,  69-72 
Thomson,  James,  isothermals,  101 
Thomson's  rule,  331 
Transition  points,  275,  277,  278 
Transport  numbers,  293,  294 
Trigonometrical  functions,  350 
Trouton's  rule,  147 

Uhler,  absorption  spectra,  208 
Units,  electrical,  5 

mechanical,  4 

thermal,  4 
Universal  wax,  78 
Uranium,  disintegration,  343 

Valson,  moduli,  159 
Van  der  Waals'  equation,  97 
applications,  100 
24 


Van  der  Waals'  constants,  355 

reduced  equation,  104-107 
Van  Dijk,  value  of  faraday,  280 
Van't  Hoff,  coefficient,  167 

equation,  167 

equilibrium  and  temperature,  250 

isochore  equation,  253 

order  of  reactions,  240 

solid  solutions,  178 

study  of  osmotic  pressure,    164- 

167 

Vapor  density,  Dumas,  91-92 
Victor  Meyer,  93 
mean,  112 
Vapor    pressure,  and    boiling   points, 

measurement  of,  101,  in 
of  alcohols,  358 
of  benzol,  359 
of  carbon  bisulphide,  359 
of  liquid  mixtures,   153-158 
of  mercury,  357 
of  solutions,  170 
of  water,  356 

Vaporization,  heat  of,  147,  263 
Vapors,  99-110 
Velocity  constant,  227 
and  mass,  342 

of  ions,  286,  291,  294,  296,  298 
of  light,  210 
of  reactions,  226,  246 
of  sound,  128 
Ventske  unit,  223 
Verdet's  constant,  220 
Vernier,  use  of,  17 

caliper,  17 
Viscosity,  118,  121 

measurement  of,  120 
Volt,  definition,  7 
Voltameter,  copper,  64-66,  282,  283 

silver,  282 
Volumenometer,  83,  84 

Walden,  non-aqueous  solutions,  307 
Walker,  diazonium  salt,  234 

electrical  measure  of  saponifica- 
tion,  300 

vapor  pressure,  36 
Welter's  rule,  194 
Wanner,  pyrometer,  202 
Warburg,  absorption  spectra,  207 

solid  electrolytes,  310 
Washburn,  hydrates,  306 
Water,  boiling  point,  33 

density,  352 

dissociation  of,  255,  327,  328 


370  INDEX. 

Water,  equivalent,  39-41  Wheatstone's  bridge,  box  type,  52 

phases,  270  Callender  and  Griffith's,  53 

specific  heat,  5  slide  wire,  50 

Watt,  definition,  4,  7  Wien,  radiation,  200,  201 

Wave  lengths,  of  ether  waves,  208  Wilhelmy,  inversion  of  sugar,  23 1 

of  light,  361  Witt  stirrer,  72 

Wax,  universal,  78  Wolff,  standard  cells,  335 

Weighing  pipette,  25  Wood,  radiation,  199 

Weston  cell,  335 

Westphal  balance,  25  Zemplen,  surface  tension,  116 

Whetham,  velocity  of  ions,  294 


RETURN     CIRCULATION  DEPARTMENT 

TO—  •*      202  Main  Library 

LOAN  PERIOD  1 
HOME  USE 

2 

3 

4 

5 

6 

ALL  BOOKS  MAY  BE  RECALLED  AFTER  7  DAYS 

Renewals  and  Recharges  may  be  made  4  days  prior  to  the  due  date. 

Books  may  be  Renewed  by  calling     642-3405. 


DUE  AS  STAMPED  BELOW 

JUL  07  1989 

UNIVERSITY  OF  CALIFORNIA,  BERKELEY 
FORM  NO.  DD6  BERKELEY,  CA  94720 

®s 


YC 


u.c. 


C°0fe7-Ul37' 


